##### Question 1:

Prove that the function f (x) = 5x – 3 is continuous at x = 0, at x = – 3 and at x = 5.

##### Answer:

##### Question 2:

Examine the continuity of the function f (x) = 2x^{2} – 1 at x = 3.

##### Answer:

##### Question 3:

Examine the following functions for continuity :

f (x) = x – 5

##### Answer:

Since f (x) = x – 5 is a polynomial,

$\therefore $ ‘f ’ is continuous at each x $\in $ R.

##### Question 4:

Examine the following functions for continuity :

##### Answer:

##### Question 5:

Examine the following functions for continuity :

##### Answer:

##### Question 6:

Examine the following functions for continuity :

f (x) = | x – 5|.

##### Answer:

##### Question 7:

Prove that the function f (x) = x^{n} is continuous at x = n, where n is a positive integer.

##### Answer:

Since f (x) = x^{n} is a polynomial,

$\therefore $ it is continuous at all n $\in $ R.

Hence ‘f ’ is continuous at x = n $\in $ N.

##### Question 8:

Is the function ‘f ’ defined by :

continuous at x = 0 ? At x = 1 ? At x = 2 ?

##### Answer:

##### Question 9:

Find all points of discontinuity of ‘f ’, where ‘f ’ is defined by

##### Answer:

At x = 2 :

##### Question 10:

Find all points of discontinuity of ‘f ’, where ‘f ’ is defined by

##### Answer:

##### Question 11:

Find all points of discontinuity of ‘f ’, where ‘f ’ is defined by

##### Answer:

At x = 0 :

##### Question 12:

Find all points of discontinuity of ‘f ’, where ‘f ’ is defined by

##### Answer:

At x = 0 :

##### Question 13:

Find all points of discontinuity of ‘f ’, where ‘f ’ is defined by

##### Answer:

At x = 1 :

##### Question 14:

Find all points of discontinuity of ‘f ’, where ‘f ’ is defined by

##### Answer:

At x = 2 :

##### Question 15:

Find all points of discontinuity of ‘f ’, where ‘f ’ is defined by

##### Answer:

At x = 1 :

##### Question 16:

Is the function defined by :

a continuous function ?

##### Answer:

At x = 1.

##### Question 17:

Discuss the continuity of the function ‘f ’, where ‘f ’ is defined by

##### Answer:

##### Question 18:

Discuss the continuity of the function ‘f ’, where ‘f ’ is defined by

##### Answer:

##### Question 19:

Discuss the continuity of the function ‘f ’, where ‘f ’ is defined by

##### Answer:

Here D_{f} = R.

Here five cases arise :

Case I. Where c < –1.

##### Question 20:

Find the relationship between ‘a’ and ‘b’ so that the function ‘f ’ defined by :

is continuous at x = 3.

##### Answer:

##### Question 21:

For what value of $\mathrm{\lambda}$ is the function :

What about continuity at x = 1 ?

##### Answer:

At x = 0 :

##### Question 22:

Show that the function defined by g (x) = x – [x] is discontinuous at all integral points.

##### Answer:

Let n $\in $ I.

##### Question 22:

Is the function defined by

f (x) = x^{2} – sin x + 5 continuous at x = $\mathrm{\pi}$ ?

##### Answer:

We have : f (x) = x^{2} – sin x + 5.

f ($\mathrm{\pi}$) = $\mathrm{\pi}$^{2} – sin $\mathrm{\pi}$ + 5 = $\mathrm{\pi}$^{2} – 0 + 5

= $\mathrm{\pi}$^{2} + 5.

= $\mathrm{\pi}$^{2} – sin $\mathrm{\pi}$ + 5

=$\mathrm{\pi}$^{2} – 0 + 5 = $\mathrm{\pi}$^{2} + 5 = f ($\mathrm{\pi}$).

Hence, ‘f’ is continuous at x = $\mathrm{\pi}$.

##### Question 24:

Discuss the continuity of the following functions :

(a) f (x) = sin x + cos x

(b) f (x) = sin x – cos x

(c) f (x) = sin x . cos x.

##### Answer:

(a) We have :

f (x) = sin + cos x, Df = R.

Let c $\in $ D_{f}.

= sin c + cos c = f (c)

$\Rightarrow $ ‘f’ is continuous at x = c.

But c is arbitrary.

Hence, ‘f’ is a continuous function.

(b) Replace (+) by (–).

(c) Replace (+) by (.).

##### Question 25:

Discuss the continuity of the cosine, cosecant, secant and cotangent functions.

##### Answer:

##### Question 26:

Find all points of discontinuity of f, where :

##### Answer:

At x = 0 :

##### Question 27:

Determine if ‘f’ defined by :

is a continuous function.

##### Answer:

At x = 0 :

##### Question 28:

Examine the continuity of ‘f ’, where ‘f ’ is defined by :

##### Answer:

At x = 0 :

##### Question 29:

Find the values of k so that the function ‘f’ is continuous at the indicated point in

##### Answer:

##### Question 30:

Find the values of k so that the function ‘f’ is continuous at the indicated point in

##### Answer:

##### Question 31:

Find the values of k so that the function ‘f’ is continuous at the indicated point in

##### Answer:

##### Question 32:

Find the values of k so that the function ‘f’ is continuous at the indicated point in

##### Answer:

##### Question 33:

Find the values of a and b such that the function defined by :

is a continuous function.

##### Answer:

Since f is continuous at all x,

$\therefore $ f is continuous at x = 2, 10.

At x = 2 :

##### Question 34:

Show that the function defined by f (x) = cos (x^{2}) is a continuous function.

##### Answer:

We have : f (x) = cos (x^{2}).

D_{f}= R.

Let c $\in $ D_{f}, arbitrary.

$\Rightarrow $ ‘f’ is continuous at x = c

$\Rightarrow $ ‘f’ is continuous on R.

##### Question 35:

Show that the function defined by f (x) = |cos x| is a continuous function.

##### Answer:

We have : f (x) = |cosx|.

D_{f} = R.

Let c $\in $ D_{f}, arbitrary.

$\Rightarrow $ ‘f’ is continuous at x = c

$\Rightarrow $ ‘f’ is continuous on R.

##### Question 36:

Show that sin |x| is a continuous function.

##### Answer:

We have : f (x) = sin |x|.

Let c be an arbitrary number $\in $ D_{f}.

##### Question 37:

Find all the points of discontinuity of ‘f’ defined by f (x) = |x| – |x + 1|.

##### Answer:

We have : f (x) = |x| – |x + 1|.

$\Rightarrow $ ‘f’ is continuous at x = 0

Also ‘f ’ being constant, is continuous when x < –1 or

when x > 0.

Thus ‘f’ is continuous for all x $\in $ R.

Hence, there is no point of discontinuity.

##### Question 38:

Differentiate the following functions with respect to x sin (x^{2} + 5)

##### Answer:

Let y = sin (x^{2} + 5).

Put x^{2} + 5 = t.

$\therefore $ y = sin t, where t = x^{2} + 5.

= cos t. (2x + 0) = 2x cos (x^{2} + 5).

##### Question 39:

Differentiate the following functions with respect to x cos (sin x)

##### Answer:

Let y = cos (sin x).

Put sin x = t.

$\therefore $ y = cos t, where t = sin x.

= (– sin t).(cos x)

= – [sin (sin x)] cos x.

##### Question 40:

Differentiate the following functions with respect to x sin (ax + b)

##### Answer:

Let y = sin (ax + b).

Put ax + b = t.

$\therefore $ y = sin t, where t = ax + b.

= a cos (ax + b).

##### Question 41:

Differentiate the following functions with respect to x

##### Answer:

##### Question 42:

Differentiate the following functions with respect to x

##### Answer:

##### Question 43:

Differentiate the following functions with respect to x
cos x^{3}. sin^{2} (x^{5})

##### Answer:

Let y = cos x^{3}.sin^{2} (x^{5}) = uν,

where u = cos x^{3} and ν = sin^{2} (x^{5}).

##### Question 44:

Differentiate the following functions with respect to x

##### Answer:

##### Question 45:

Differentiate the following functions with respect to x

##### Answer:

##### Question 46:

Prove the function ‘f’ given by :

f (x) = |x – 1|, x $\in $ R

is not differentiable at x = 1.

##### Answer:

##### Question 47:

Prove that the greatest integer function defined by [x], 0 < x < 3 is not differentiable at x = 1 and x = 2.

##### Answer:

Let f (x) = [x].

##### Question 48:

2x + 3y = sin x

##### Answer:

We have : 2x + 3y = sin x.

##### Question 49:

2x + 3y = sin y

##### Answer:

We have : 2x + 3y = sin y.

##### Question 50:

ax + by^{2} = cos y

##### Answer:

We have : ax + by^{2} = cos y.

##### Question 51:

xy + y^{2} = tan x + y

##### Answer:

We have : xy + y^{2} = tan x + y.

##### Question 52:

x^{2} + xy + y^{2 = 100}

##### Answer:

We have : x^{2} + xy + y^{2} = 100.

##### Question 53:

x^{3} + x^{2}y + xy^{2} + y^{3} = 81

##### Answer:

We have : x^{3} + x^{2}y + xy^{2} + y^{3} = 81.

##### Question 54:

sin^{2} y + cos xy = $\mathrm{\pi}$

##### Answer:

We have : sin^{2} y + cos xy = $\mathrm{\pi}$.

Diff. w.r.t. x,

##### Question 55:

sin^{2} x + cos^{2} y = 1.

##### Answer:

We have : sin^{2} x + cos^{2} y = 1.

Diff. w.r.t. x,

##### Question 56:

##### Answer:

##### Question 57:

##### Answer:

##### Question 58:

##### Answer:

##### Question 59:

##### Answer:

##### Question 60:

##### Answer:

##### Question 61:

##### Answer:

##### Question 62:

##### Answer:

##### Question 63:

Differentiate the following w.r.t. x :

##### Answer:

##### Question 64:

Differentiate the following w.r.t. x :

##### Answer:

##### Question 65:

Differentiate the following w.r.t. x :

##### Answer:

##### Question 66:

Differentiate the following w.r.t. x :

sin (tan^{–1} e^{–x}

##### Answer:

##### Question 67:

Differentiate the following w.r.t. x :

log (cos e^{x})

##### Answer:

##### Question 68:

Differentiate the following w.r.t. x :

##### Answer:

##### Question 69:

Differentiate the following w.r.t. x :

##### Answer:

##### Question 70:

Differentiate the following w.r.t. x :

log (log x), x > 1

##### Answer:

##### Question 71:

Differentiate the following w.r.t. x :

##### Answer:

##### Question 72:

Differentiate the following w.r.t. x :

cos (log x + e^{x}), x > 0.

##### Answer:

##### Question 73:

Differentiate the functions given in w.r.t. x :

cos x. cos 2x. cos 3x.

##### Answer:

Let y = cos x.cos 2x.cos 3x.

Taking logs.,

log y = log (cos x. cos 2x.cos 3x) ...(1)

= log cos x + log cos 2x + log cos 3x.

##### Question 74:

Differentiate the functions given in w.r.t. x :

##### Answer:

##### Question 75:

Differentiate the functions given in w.r.t. x :

(log x)cos x.

##### Answer:

##### Question 76:

Differentiate the functions given in w.r.t. x :

xx – 2sin x

##### Answer:

##### Question 77:

Differentiate the functions given in w.r.t. x :

(x + 3)2.(x + 4)3.(x + 5)4.

##### Answer:

Let y = (x + 3)^{2} (x + 4)^{3} (x + 5)^{4} ...(1)

Taking logs., log y = log (x + 3)^{2} (x + 4)^{3}(x + 5)^{4}

= 2 log (x + 3) + 3 log (x + 4) + 4 log (x + 5).

##### Question 78:

Differentiate the functions given in w.r.t. x :

##### Answer:

##### Question 79:

Differentiate the functions given in w.r.t. x :

(log x )^{x} + xlog x

##### Answer:

##### Question 80:

Differentiate the functions given in w.r.t. x :

##### Answer:

##### Question 81:

Differentiate the functions given in w.r.t. x :

xsin x + (sin x)cosx

##### Answer:

##### Question 82:

Differentiate the functions given in w.r.t. x :

##### Answer:

##### Question 83:

Differentiate the functions given in w.r.t. x :

##### Answer:

##### Question 84:

x^{y} + y^{x} = 1.

##### Answer:

We have : x^{y} + y^{x} = 1.

Putting x^{y} = u and y^{x} = ν, we get :

u + ν= 1

##### Question 85:

y^{x} = x^{y}.

##### Answer:

##### Question 86:

(cos x)^{y} = (cos y)^{x}.

##### Answer:

##### Question 87:

xy = e^{(x – y)}.

##### Answer:

##### Question 88:

Find the derivative of the function given by :

f (x) = (1 + x) (1 + x^{2}) (1 + x^{4}) (1 + x^{8}) and hence find f ′ (1).

##### Answer:

(i) We have :

f(x) = (1 + x) (1 + x^{2}) (1 + x^{4}) (1 + x^{8}) ...(1)

Taking logs., log f(x)

= log (1 + x) + log (1 + x^{2}) + log (1 + x^{4}) + log (1 + x^{8}).

##### Question 89:

Differentiate (x^{2} – 5x + 8) (x^{3} + 7x + 9) in three ways

mentioned below :

(i) by using product rule

(ii) by expanding the product to obtain a single polynomial

(iii) by logarithmic differentiation.

Do they give the same answer ?

##### Answer:

We have :

f(x) = (x^{2} – 5x + 8) (x^{3} + 7x + 9) ...(1)

(i) By Product Rule :

= (x^{2} – 5x + 8) (3x^{2} + 7)

+ (x^{3} + 7x + 9) (2x – 5)

= 3x^{4} –15x^{3} + 24x^{2} + 7x^{2} – 35x + 56 + 2x^{4}

+ 14x^{2} + 18x – 5x^{3} – 35x – 45

= 5x^{4} – 20x^{3} + 45x^{2} – 52x + 11.

(ii) By Expansion :

f(x) = x^{5} + 7x^{3} + 9x^{2} – 5x^{4} – 35x^{2} – 45x

+ 8x^{3} + 56x + 72

= x^{5} – 5x^{4} + 15x^{3} – 26x^{2} + 11x + 72.

$\therefore $ f′(x) = 5x^{4} – 20x^{3} + 45x^{2} – 52x + 11.

(iii) By Logarithmic Differentiation :

Taking logs., on both sides of (1),

log f (x) = log (x^{2} – 5x + 8) (x^{3} + 7x + 9)

= log (x^{2} – 5x + 8) + log (x^{3} + 7x + 9).

= (2x – 5) (x^{3} + 7x + 8) + (3x^{2} + 7) (x^{2} – 5x + 8)

= 5x^{4} – 20x^{3} + 45x^{2} – 52x + 11.

Hence, the three answers are the same.

##### Question 90:

If u, ν, w are differentiable functions of x, then show that :

in two ways by repeated application of product rule, second by logarithmic differentiation.

##### Answer:

##### Question 91:

If x and y are connected parametrically by the equations given below without eliminating the parameters.

x = 2at^{2}, y = at^{4}.

##### Answer:

##### Question 92:

If x and y are connected parametrically by the equations given below without eliminating the parameters.

x = a cos $\mathrm{\theta}$, y = b cos $\mathrm{\theta}$.

##### Answer:

##### Question 93:

If x and y are connected parametrically by the equations given below without eliminating the parameters.

x = sin t, y = cos 2t.

##### Answer:

##### Question 94:

##### Answer:

##### Question 95:

x = cos $\mathrm{\theta}$ – cos 2$\mathrm{\theta}$, y = sin $\mathrm{\theta}$ – sin 2$\mathrm{\theta}$.

##### Answer:

##### Question 96:

x = a ($\mathrm{\theta}$ – sin $\mathrm{\theta}$), y = a (1 + cos $\mathrm{\theta}$).

##### Answer:

##### Question 97:

##### Answer:

##### Question 98:

If x and y are connected parametrically by the equations given in without eliminating the parameters.

##### Answer:

##### Question 99:

If x and y are connected parametrically by the equations given in without eliminating the parameters.

x = a sec $\mathrm{\theta}$ , y = b tan $\mathrm{\theta}$.

##### Answer:

##### Question 100:

If x and y are connected parametrically by the equations given in without eliminating the parameters.

x = a (cos $\mathrm{\theta}$ + $\mathrm{\theta}$ sin $\mathrm{\theta}$), y = a (sin $\mathrm{\theta}$ – $\mathrm{\theta}$ cos $\mathrm{\theta}$).

##### Answer:

##### Question 101:

##### Answer:

##### Question 102:

Find the second order derivative of the functions given in x^{2} + 3x + 2

##### Answer:

Let y = x^{2} + 3x + 2.

##### Question 103:

Find the second order derivative of the functions given in x^{20}

##### Answer:

Let y = x^{20}.

##### Question 104:

Find the second order derivative of the functions given in x cos x

##### Answer:

Let y = x cos x.

##### Question 105:

Find the second order derivative of the functions given in log x

##### Answer:

Let y = log x.

##### Question 106:

Find the second order derivative of the functions given in x^{3} log x

##### Answer:

Let y = x^{3} log x.

##### Question 107:

Find the second order derivative of the functions given in e^{x} sin 5x

##### Answer:

##### Question 108:

Find the second order derivative of the functions given in e^{6x} cos 3x

##### Answer:

##### Question 109:

Find the second order derivative of the functions given in tan–^{1} x

##### Answer:

##### Question 110:

Find the second order derivative of the functions given in log (log x)

##### Answer:

##### Question 111:

Find the second order derivative of the functions given in sin (log x).

##### Answer:

##### Question 112:

If = 5 cos x – 3 sin x, prove that :

##### Answer:

We have : y = 5 cos x – 3 sin x ...(1)

##### Question 113:

##### Answer:

We have :

y = cos^{–1} x

$\Rightarrow $ cos y = x ...(1)

##### Question 114:

If y = 3 cos (log x) + 4 sin (log x), show that :

x_{2}y_{2} + y_{1} + y = 0.

##### Answer:

$\Rightarrow $ x^{2}y^{2 + xy1 = – (3 cos (log x) + 4 sin (log x))
= – y. [Using (1)]
Hence, x2y2 + xy1 + y = 0.}

##### Question 115:

If y = A emx + B enx, show that :

##### Answer:

##### Question 116:

If y = 500 e7x + 600 e–7xm, show that :

##### Answer:

##### Question 117:

If ey (x + 1) = 1, show that :

##### Answer:

We have : e^{y} (x + 1) = 1

##### Question 118:

If y = (tan^{–1} x)^{2}, show that :

(x^{2} + 1)^{2} y_{2} + 2x (x^{2} + 1) y_{1} = 2.

##### Answer:

We have : y = (tan^{–1} x)^{2} ...(1)

##### Question 119:

Verify Rolle’s theorem for the function :

f (x) = x2^{ + 2x – 8, x $\in $ [– 4, 2].}

##### Answer:

We have : f (x) = x^{2} + 2x – 8.

(I) f (x) is continuous in [– 4, 2].

[$\because $ f (x) is a polynomial in x]

(II) f′ (x) = 2x + 2 ...(1)

$\therefore $ f′ (x) exists for each x $\in $ (– 4, 2).

(III) f (– 4) = (– 4^{)2} + 2 (– 4) – 8

= 16 – 8 – 8 = 0,

f (2) = (2)^{2} + 2 (2) – 8

= 4 + 4 – 8 = 0.

$\therefore $ f (– 4) = f (2).

Thus all the conditions of Rolle’s Theorem are satisfied.

$\therefore $ There exists at least one number ‘c’ between – 4 and 2

s.t. f′ (c) = 0.

But f′ (c) =2c + 2 [Putting x = c in (1)]

$\therefore $ f′ (c) = 0 gives 2c + 2 = 0

$\Rightarrow $ c = – 1 $\in $ (– 4, 2).

Hence, the theorem is verified and c = – 1.

##### Question 120:

Examine if Rolle’s theorem is applicable to any of the following functions. Can you say something about the
converse of Rolle’s theorem from these examples ?

(i) f (x) = [x] for x $\in $ [5, 9]

(ii) f (x) = [x] for x $\in $ [– 2, 2]

(iii) f (x) = x2 – 1 for x $\in $ [1, 2].

##### Answer:

(a) (i) We have :

f (x) = [x] for x $\in $ [5, 9].

f(x) is neither continuous nor derivable at x = 6, 7, 8.

Hence, Rolle’s Theorem is not applicable.

(ii) We have f (x) = [x] for x $\in $ [– 2, 2].

f (x) is neither continuous nor derivable at
x = – 1, 0, 1.

Hence, Rolle’s theorem is not applicable.

(iii) We have : f (x) = x^{2} – 1 for x $\in $ [1, 2].

(I) f (x) is continuous in [1, 2].

[$\because $ f (x) is a polynomial in x]

(II) f′ (x) = 2x, which exists in (1, 2)

$\Rightarrow $ f (x) is derivable in (1, 2).

(III) f (1) = 1 – 1 = 0, f (2) = 4 – 1 = 3

$\Rightarrow $ f (1) $\ne $ f (2).

Hence, Rolle’s Theorem is not applicable.

(b) Conversely :

If f′ (c) = 0, c $\in $ [a, b], the conditions of Rolle’s theorem
are not true.

##### Question 121:

If f : [– 5, 5] $\to $ R is differentiable function and if f′ (x) does not vanish anywhere, then prove that f (– 5) $\ne $ f (5).

##### Answer:

Let us assume that f (– 5) = f (5).

Then ‘f’ satisfies all the conditions of Rolle’s Theorem in
[– 5, 5].

[$\because $ Differentiability $\Rightarrow $ Continuity]

Then there must exist at least one c $\in $ (– 5, 5) such that
f′ (c) = 0.

Thus our supposition is wrong.

Hence, f (– 5) $\ne $ f (5).

##### Question 122:

Verify Mean Value Theorem. if
f (x) = x^{2} – 4x – 3 in the interval [a, b],
where a = 1 and b = 4.

##### Answer:

Solution. We have :

f (x) = x^{2} – 4x – 3 ...(1)

(I) f (x) is continuous in [1, 4].

[$\because $ f (x) is a polynomial in x]

(II) f′ (x) = 2x – 4 ...(2)

Thus f′ (x) exists for each x in (1, 4).

Thus both the conditions of Mean Value Theorem are
satisfied.

$\therefore $There exists at least one number ‘c’ between a and b

But f′(c) = 2c – 4 [Putting x = c in (2)]

f (a) = f (1) = 1 – 4 – 3 = – 6

f (b) = f (4) = 16 – 16 – 3 = – 3.

$\Rightarrow $ 2c – 4 = 1

$\Rightarrow $ 2c = 5

##### Question 123:

Verify Mean Value Theorem if :

f (x) = x^{3} – 5x^{2} – 3x in the interval [a, b], where

a = 1 and b = 3. Find all c $\in $ (1, 3) for which f ′(c) = 0.

##### Answer:

(i) We have :

f (x) = x^{3} – 5x^{2} – 3x ...(1)

(I) Since f (x) is a polynomial in x,

$\therefore $ it is continuous in [1, 3].

(II) f′ (x) = 3x^{2} – 10x – 3 ...(2),

which exists in (1, 3).

Thus both the conditions of Mean Value Theorem are
satisfied.

$\therefore $There exists at least one point ‘c’ in (1, 3) such that :

But f′ (c) = 3c^{2} – 10c – 3,

f (a) = f (1) = 1 – 5 – 3 = – 7

and f (b) = f (3) = 27 – 5 (9) – 3 (3)

= 27 – 45 – 9 = – 27.

$\Rightarrow $ 3c^{2} – 10c – 3 = – 10

$\Rightarrow $ 3c^{2} – 10c + 7 = 0.

##### Question 124:

Examine the applicability of Mean Value Theorem
for the following functions :

(i) f (x) = [x] for x $\in $ [5, 9]

(ii) f (x) = [x] for x $\in $ [– 2, 2]

(iii) f (x) = x^{2} – 1 for x $\in $ [1, 2].

##### Answer:

(i) We have : f (x) = [x] ; x $\in $ [5, 9].

Clearly f (x) is neither continuous nor derivable at integral
points of the interval [5, 9],

i.e. at x = 5, 6, 7, 8 and 9.

Thus f (x) is neither continuous in [5, 9] nor derivable in
(5, 9).

Hence, Mean Value Theorem is not applicable to f (x) in
[5, 9].

(ii) We have : f (x) = [x], x $\in $ [– 2, 2].

Clearly f (x) is neither continuous nor derivable. at
x = – 2, – 1, 0, 1, 2.

Thus f (x) is neither continuous in [– 2, 2] nor derivable
in (– 2, 2).

Hence, Mean value Theorem is not applicable to f (x) in
[– 2, 2].

(iii) We have f (x) = x^{2} – 1.

It is continuous in [1, 2].

[$\because $ f (x) is a polynomial in x]

(II) f′ (x) = 2x, which exists in (1, 2).

Thus both the conditions of Mean Value Theorem are
satisfied.

$\therefore $ There exists at least one point ‘c’ in (1, 2) such that :

##### Question 125:

continuous at x = 0, then the value of ‘k’ is:

- 3
- 2
- 1
- 1.5

##### Answer:

2

##### Question 126:

The function f(x) = [x], where [x] denotes the greatest integer function, is continuous at :

- 4
- –2
- 1
- 1.5

##### Answer:

1.5

##### Question 127:

The value of ‘k’ which makes the function defined by :

continuous at x = 0 is :

- 8
- 3
- –1
- None of these

##### Answer:

None of these

##### Question 128:

Differential coefficient of sec (tan^{–1} x) w.r.t. x is :

##### Answer:

##### Question 129:

##### Answer:

##### Question 130:

##### Answer:

##### Question 131:

- x
- 1

##### Answer:

1

##### Question 132:

##### Answer:

##### Question 133:

- 1
- -1

##### Answer:

1

##### Question 134:

The value of ‘c’ in Mean Value Theorem for the function f(x) = x (x – 2), x $\in $ [1, 2] is :