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Which of the following is the fastest mode of transport?

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Important Questions on Application of Derivatives

HARD

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

Find two positive numbers whose sum is

15

and the sum of whose squares is minimum.

HARD

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

Suppose

f (x)

is a polynomial of degree four having critical points at

- 1, 0, 1

. If

T = \{x \in R | f (x) = f (0)\}

, then the sum of squares of all the elements of

T

is :

HARD

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

An open tank with a square base and vertical sides is to be constructed from a metal sheet so as to hold a given quantity of water. Show that the cost of material will be least when depth of the tank is half of its width. If the cost is to be borne by nearby settled lower income families, for whom water will be provided, what kind of value is hidden in this question ?

EASY

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

Show that the height of the right circular cylinder of greatest volume which can be inscribed in a right circular cone of height

h

and radius

r

is one-third of the height of the cone, and the greatest volume of the cylinder is

\frac{4}{9}

times the volume of the cone.

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

f (x)

is a non-zero polynomial of degree four, having local extreme points at

x = - 1, 0, 1;

then the set

S = {x \in R : f (x) = f (0)}

contains exactly

HARD

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

A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is $2 m$ and volume is $8 m^{3}$ . If building of tank costs $₹ 70$ per square metre for the base and $₹ 45$ per square metre for the sides, what is the cost of least expensive tank $?$

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 :

HARD

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

The point on the curve

x^{2} = 2 y

which is nearest to the point

(0, 5)

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

S_{1}

and

S_{2}

are respectively the sets of local minimum and local maximum points of the function,

f (x) = 9 x^{4} + 12 x^{3} - 36 x^{2} + 25, x \in R,

then

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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:

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

A $10 V$ cell of negligible internal resistance is connected in parallel across a battery of emf $200 V$ and internal resistance $38 Ω$ as shown in the figure. Find the value of current in the circuit.

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