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Let the tangent to the graph of $y = f (x)$ at the point $x = a$ be parallel to $x$ -axis and let $f^{'} (a - h) > 0$ and $f^{'} (a + h) < 0$ where $h$ is a very small $+ ve$ quantity. Then, the ordinate at $x = a$ is -

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Important Questions on Application of Derivatives

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

x = - 1

and

x = 2

are extreme points of

f (x) = α \log |x| + β x^{2} + x

, then

HARD

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

If a right circular cone, having maximum volume, is inscribed in a sphere of radius

3 c m

, then the curved surface area (in

{cm}^{2}

) of this cone is :

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

From the top of a

64

metres high tower, a stone is thrown upwards vertically with the velocity of

48 m / s .

The greatest height (in metres) attained by the stone, assuming the value of the gravitational acceleration

g = 32 m / s^{2}

, is:

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

The maximum volume

(in cubic m)

of the right circular cone having slant height

3 m

is:

EASY

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

f (x) = \frac{x}{x^{2} + 1}

is an increasing function then the value of

x

lies in

HARD

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

The maximum area of a rectangle that can be inscribed in a circle of radius

2

units is

HARD

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

Let

k

and

K

be the minimum and the maximum values of the function

f (x) = \frac{{(1 + x)}^{0.6}}{1 + x^{0.6}} in [0, 1]

, respectively, then the ordered pair

(k, K)

is equal to:

EASY

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

The maximum value of the function

f (x) = 2 x^{3} - 15 x^{2} + 36 x + 4

is attained at

HARD

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

The maximum area (in sq. units) of a rectangle having its base on the

x -

axis and its other two vertices on the parabola,

y = 12 - x^{2}

such that the rectangle lies inside the parabola, is :

HARD

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

If the function

f

given by

f (x) = x^{3} - 3 (a - 2) x^{2} + 3 a x + 7

, for some

a \in R

is increasing in

(0, 1]

and decreasing in

[1, 5)

, then a root of the equation,

\frac{f (x) - 14}{{(x - 1)}^{2}} = 0, (x \neq 1)

is :

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

Twenty meters of wire is available for fencing off a flower-bed in the form of a circular sector. Then the maximum area (in sq. m) of the flower-bed, is:

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

The difference between the greatest and the least value of

f (x) = 2 \sin x + \sin 2 x, x \in [0, \frac{3 π}{2}]

HARD

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

The maximum value of

f (x) = \frac{\log x}{x} (x \neq 0, x \neq 1)

HARD

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

A solid hemisphere is mounted on a solid cylinder, both having equal radii. If the whole solid is to have a fixed surface area and the maximum possible volume, then the ratio of the height of the cylinder to the common radius is

EASY

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

Let

M

and

m

be respectively the absolute maximum and the absolute minimum values of the function,

f (x) = 2 x^{3} - 9 x^{2} + 12 x + 5

in the interval

[0, 3]

. Then

M - m

is equal to

HARD

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

Among all sectors of a fixed perimeter, choose the one with maximum area. Then the angle at the center of this sector (i.e., the angle between the bounding radii) is-

HARD

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

Let

f (x) = α x^{2} - 2 + \frac{1}{x}

where

α

is a real constant. The smallest

α

for which

f (x) \geq 0

for all

x > 0

is-

HARD

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

The maximum value of

f (x) = \frac{x}{4 + x + x^{2}}

[- 1, 1]

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

The least value of

α \in R

for which,

4 α x^{2} + \frac{1}{x} \geq 1,

for all

x > 0,

EASY

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

If at

x = 1,

the function

x^{4} - 62 x^{2} + a x + 9

attains its local maximum value, on the interval

[0, 2]

, then the value of

a

Let the tangent to the graph of y =fx at the point x =a be parallel to x -axis and let f'a- h>0 and f'a + h < 0 where h is a very small +ve quantity. Then, the ordinate at x =a is -

Important Questions on Application of Derivatives

Learn and Prepare for any exam you want !

Let the tangent to the graph of $y = f (x)$ at the point $x = a$ be parallel to $x$ -axis and let $f^{'} (a - h) > 0$ and $f^{'} (a + h) < 0$ where $h$ is a very small $+ ve$ quantity. Then, the ordinate at $x = a$ is -