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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Let $f : (0, 1) \to ℝ$ be the function defined as $f (x) = [4 x] {(x - \frac{1}{4})}^{2} (x - \frac{1}{2})$ , where $[x]$ denotes the greatest integer less than or equal to $x$ . Then which of the following is(are) true?

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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 :

Let f:0,1→ℝ be the function defined as fx=4xx-142x-12, where [x] denotes the greatest integer less than or equal to x. Then which of the following is(are) true?

Important Questions on Application of Derivatives

Learn and Prepare for any exam you want !

Let $f : (0, 1) \to ℝ$ be the function defined as $f (x) = [4 x] {(x - \frac{1}{4})}^{2} (x - \frac{1}{2})$ , where $[x]$ denotes the greatest integer less than or equal to $x$ . Then which of the following is(are) true?