Calculus

Application of Derivatives

Medium

core

A cylinder with height 'h' and radius of base is 'r' which is open at the top having a given outer surface area, then volume of the cylinder is maximum if

$r=2h$

$h=3r$

$h=r$

$h=2r$

✅ Correct Option: 3

With the top open, $S=\pi r^2+2\pi rh$ is fixed, so $h=\dfrac{S-\pi r^2}{2\pi r}$ and $V=\pi r^2h=\dfrac{Sr-\pi r^3}{2}$ . Setting $\dfrac{dV}{dr}=\dfrac{S-3\pi r^2}{2}=0$ gives $S=3\pi r^2$ . Substituting back, $2\pi rh=2\pi r^2$ , so $h=r$ .

2025: 15 May Shift 1

Which of the following are correct?

(A) The function $f(x) = 3x+12$ is increasing on R.

(B) The function $f(x) = e^{2x}$ is decreasing on R.

(D) The function $f(x) = x^3-3x^2+4x$ is increasing on R.

Choose the correct answer from the options given below:

2026: 18 May Shift 1

The function $f(x) = 2x^3-15x^2+36x+4$ has local maxima at $x=$

2026: 22 May Shift 2

The function $f(x)=x^2-x\sqrt{2}$ is strictly decreasing in the interval :

2026: 31 May Shift 2

The ordinate of the point on the curve $y^2 = 8x$ for which abscissa and ordinate change at the same rate is:

2025: 15 May Shift 1

The function $f(x) = x^2 - 4x + 6$ is

(A) Strictly decreasing on $(-\infty, 2) \cup (2, \infty)$

(B) Strictly increasing on $(2, \infty)$

(D) Strictly decreasing on $(-\infty, 2)$

Choose the correct answer from the options given below:

2026: 31 May Shift 2

If the function $f(x) = x^4 - 62x^2 + ax + 9$ has a critical point at $x = 1$ , then $a$ is equal to:

A cylinder with height 'h' and radius of base is 'r' which is open at the top having a given outer surface area, then volume of the cylinder is maximum if

Solution

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