Find the area of the largest rectangle with lower base on the x -axis and upper vertices on the curve y = 12 - x 2 .

Area of rectangle P lies on the parabola and y = 12 - x 2 , so P = P x , 12 - x 2 Due to symmetry, the width of the rectangle is half the distance between P and the y -axis, i.e. width = 2 x and height = y Hence, the area of the rectangle is A = Width × Height ∴ A = 2 x y A = 2 x 12 - x 2 A = 24 x - 2 x 3 . . . i We are asked to maximise the area as x changes, so hopefully we can identify a critical point of A associated with a maximum. So we need to find d A d x . Differentiating i w.r.t. x ∴ d A d x = 24 - 6 x 2 . . . ii At a critical point, d A d x = 0 ∴ 24 - 6 x 2 = 0 ⇒ 6 x 2 = 24 ⇒ x 2 = 4 ∴ x = ± 2 Obviously x must be positive (otherwise we have an imaginary rectangle with negative area for a box that has collapsed in on itself) ∴ x = 2 We need to check if this is a maximum or a minimum, so differentiate ii w.r.t. x to get ∴ d 2 A d x 2 = - 12 x d 2 A d x 2 = - 12 x < 0 when x = 2 , confirming a maximum When x = 2 we have: Width = 2 · 2 = 4 Height = 12 - 4 = 8 Area = 32 sq. units

The combined resistance R of two resistors R 1 and R 2 , where R 1 , R 2 > 0 is given by 1 R = 1 R 1 + 1 R 2 . If R 1 + R 2 = C (constant), then show that the maximum resistance R is obtained by choosing R 1 = R 2 .

Here, 1 R = 1 R 1 + 1 R 2 ⇒ 1 R = R 1 + R 2 R 1 R 2 ⇒ 1 R = C R 1 C - R 1 ⇒ R = R 1 C - R 1 C ⇒ R = 1 C C R 1 - R 1 2 ⇒ d R d R 1 = 1 C C - 2 R 1 d R d R 1 = 0 ⇒ 1 C C - 2 R 1 = 0 ⇒ R 1 = C 2 d 2 R d R 1 2 = 1 C [ - 2 ] = - 2 C At R 1 = C 2 , d 2 R d R 2 = - 2 C < 0 ⇒ R has a local maxima at R 1 = C 2 But R 1 = C 2 is the only extreme point. ∴ R has maximum when R 1 = C 2 , R 2 = C - C 2 i . e when R 1 = R 2

E M B I B E

Alpha Question Bank for Engineering: Mathematics>Chapter 28 - Application of Derivatives>Exercise-1>Q 62

HARD

JEE Main/Advance

IMPORTANT

Earn 100

John has $x$ children by his first wife and Mary has $x + 1$ children by her first husband. They marry and have children of their own. The whole family has $24$ children. It is given that the children of the same parents don't fight. Then find the maximum number of fights that can take place in the family.

Important Questions on Application of Derivatives

HARD

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 28 - Application of Derivatives>Exercise-1>Q 63

If the sum of the lengths of the hypotenuse and a side of a right angled triangle is given, show that the area of the triangle is maximum when the angle between them is

\frac{π}{3}

.

HARD

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 28 - Application of Derivatives>Exercise-1>Q 64

Find the volume of the largest cylinder that can be inscribed in a sphere of radius

r cm .

HARD

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 28 - Application of Derivatives>Exercise-1>Q 65

Show that the semi-vertical angle of a right circular cone of maximum volume, of a given slant height, is

\tan^{- 1} \sqrt{2}

.

MEDIUM

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 28 - Application of Derivatives>Exercise-1>Q 66

A running track of

440

yards is to be laid out encircling a football field the shape of which is rectangle with a semi-circle at each end. If the area of the rectangular position is to be maximum find the lengths of its sides.

HARD

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 28 - Application of Derivatives>Exercise-1>Q 67

Find the area of the largest rectangle with lower base on the