For every pair of continuous functions f , g : 0 , 1 → R such that max f x : x ∈ 0 , 1 = max { g x : x ∈ 0 , 1 } , then the correct statement(s) is(are)

f c 2 = g c 2 for some c ∈ 0 , 1

f c 2 + f c = g c 2 + 3 g c for some c ∈ 0 , 1

f c 2 + 3 f c = g c 2 + g c for some c ∈ 0 , 1

HARD

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A swimmer $S$ is in the sea at a distance of $d km$ from the closest point $A$ on a straight shore. The house of the swimmer is on the shore at a distance of $L km$ from $A$ . He can swim at speed of $u {kmhr}^{- 1}$ and walk at speed of $v {kmhr}^{- 1}$ $(v > u)$ . If $x$ is the distance of the point on the shore from $A$ where he must land to reach his house in the shortest time (in $km$ ), $t$ is minimum time taken to reach his house (in $hr$ ). Then

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Important Questions on Application of Derivatives

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 :

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:

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

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,

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:

HARD

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

The function

f (x) = 2 |x| + |x + 2| - ||x + 2| - 2 |x||

has a local minimum or a local maximum at

x =

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)

MEDIUM

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

If non-zero real numbers

b

and

c

are such that

m i n f (x) > m a x g (x)

, where

f (x) = x^{2} + 2 b x + 2 c^{2}

and

g (x) = - x^{2} - 2 c x + b^{2}, (x \in R)

; then

|\frac{c}{b}|

lies in the interval

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

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

HARD

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

Let

f : R \to (0, \infty)

and

g : R \to R

be twice differentiable functions such that

f^{″}

and

g^{″}

are continuous functions on

R

. Suppose

f^{'} (2) = g (2) = 0, f^{″} (2) \neq 0

and

g^{'} (2) \neq 0

\lim_{x \to 2} \frac{f (x) g (x)}{f^{'} (x) g^{'} (x)} = 1

, then

HARD

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

The maximum value of

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

[- 1, 1]

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

A rectangular sheet of fixed perimeter with sides having their lengths in the ratio $8 : 15$ is converted into an open rectangular box by folding after removing squares of equal area from all four corners. If the total area of removed squares is $100$ , the resulting box has maximum volume. Then the lengths of the sides of the rectangular sheet are:

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

For every pair of continuous functions

f, g : [0, 1] \to R

such that

\max \{f (x) : x \in [0, 1]\}

= \max {g (x) : x \in [0, 1]}

, then the correct statement(s) is(are)

HARD

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

Let

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

and

g (x) = x - \frac{1}{x}, x \in R - \{- 1, 0, 1\} .

h (x) = \frac{f (x)}{g (x)}

, then the local minimum value of

h (x)

is:

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

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:

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 :

Important Questions on Application of Derivatives

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