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John has $x$ children by his first wife. Mary has $(x + 1)$ children by her first husband. They marry and have children of their own. The whole family has $24$ children. Assuming the two children of same parents do not fight, then find the maximum possible number of fights that can take place

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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 :

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:

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

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:

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

John has x children by his first wife. Mary has x+1 children by her first husband. They marry and have children of their own. The whole family has 24 children. Assuming the two children of same parents do not fight, then find the maximum possible number of fights that can take place

Important Questions on Application of Derivatives

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John has $x$ children by his first wife. Mary has $(x + 1)$ children by her first husband. They marry and have children of their own. The whole family has $24$ children. Assuming the two children of same parents do not fight, then find the maximum possible number of fights that can take place