The height of a closed cylinder of given surface area and maximum volume is equal to the Option 1: Radius of the base Option 2: Half of radius of the base Option 3: Two third of radius of the base Option 4: Double the radius of the base

Answer: Double the radius of the base. Solution: With S = 2 r^2 + 2 rh fixed, write h in terms of r and maximise V = r^2 h. Setting dV/dr = 0 gives S = 6 r^2, which on substitution yields h = 2r, i.e. double the base radius.

Calculus

Application of Derivatives

Medium

core

The height of a closed cylinder of given surface area and maximum volume is equal to the

Radius of the base

Half of radius of the base

Two third of radius of the base

Double the radius of the base

✅ Correct Option: 4

With $S = 2\pi r^2 + 2\pi rh$ fixed, write $h$ in terms of $r$ and maximise $V = \pi r^2 h$ . Setting $dV/dr = 0$ gives $S = 6\pi r^2$ , which on substitution yields $h = 2r$ , i.e. double the base radius.

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The maximum value of $x^{-x}$ is

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If it is given that at $x = 1$ , the function $f(x) = x^4 - 62x^2 + 2ax + b$ attains its maximum value on the interval [0, 2], then the value of a is:

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For the function $f(x) = e^x + e^{-x}$

(A) $f'(x) = e^x - e^{-x}$

(B) The critical point is $x = 0$

(D) $x = 0$ is the point of local minimum.

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The rate of change of volume of a sphere with respect to its surface area, when the radius is 6cm is:

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Match List-I with List-II

List-I	List-II
(A) Marginal average cost if cost function $C(x) = \frac{50}{\sqrt{x}}$	(I) $50\sqrt{x}$
(B) Marginal average cost if cost function $C(x) = 50\sqrt{x}$	(II) $-\frac{75}{x^2\sqrt{x}}$
(C) Revenue function if demand function $P=\frac{50}{\sqrt{x}}$	(III) $\frac{-25}{x\sqrt{x}}$
(D) Marginal revenue if demand function $P=50\sqrt{x}$	(IV) $75\sqrt{x}$

Choose the correct answer from the options given below:

The height of a closed cylinder of given surface area and maximum volume is equal to the

Solution

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