CUET Mathematics: CalculusApplication of Derivatives. Free, no login required.

The volume of spherical balloon is increasing at the rate of $4 \text{ cm}^3/ \text{sec}$. The rate of increase of its surface area, when the radius is 3cm will be :-

The interval(s), where the function $f(x) = \begin{cases} \frac{1-e^x}{e^{2x}-1} & : x \neq 0 \\ \frac{-1}{2} & : x = 0 \end{cases}$ is increasing, is/ are:

A square board of side 36cm is made into a box without top by cutting a square from each corner and folding up the flaps to form a box then maximum volume of the box is

The largest interval, in which the function $f(x) = x^3 + 2x^2 - 1$ is increasing, is:

The maximum value of $\left(\frac{1}{x}\right)^x$ for $x > 0$ is

A boat 10 m high floating at a uniform speed of 13 meters per minute(m/min) away from a lamp post 15 m high. Then the rate at which the length of shadow of the boat increases is:

If the function $f(x) = 2x^2 - kx + 7$, is increasing on $[1,2]$, then $k$ lies in the interval

Which of the following statements are true?

(A) The function $f(x) = \frac{x^4}{4} - \frac{4}{3}x^3 + \frac{x^2}{2} + 6x$ has 3 critical points.

(B) The function $f(x) = |x| + 3$ has no minimum value.

(C) A local maximum value is always the absolute maximum value.

(D) $f(x) = x^2$ has minima at $x=0$.

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The rate of change of the area of a circle with respect to its radius $r$, when $r = 3$cm, is:

The curve $x = y^2$ and $xy = k$ cut orthogonally, then $k^2$ is equal to:

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The function $f(x) = x^2e^{-2x}$ increases on

The function $f(x) = \frac{-3}{4}x^4 - 8x^3 - \frac{45}{2}x^2 + 163$ has a local maxima at

The rate of change of volume of a sphere with respect to its surface area, when radius is 4 cm, is equal to

The function $f(x) = \log_e(\sin x), x \in (0, \pi)$ is

(A) strictly increasing on $\left(0, \frac{\pi}{2}\right)$

(B) strictly decreasing on $\left(0, \frac{\pi}{2}\right)$

(C) strictly increasing on $\left(\frac{\pi}{2}, \pi\right)$

(D) strictly decreasing on $\left(\frac{\pi}{2}, \pi\right)$

(E) strictly increasing on $(0, \pi)$

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The demand function for a certain product is represented by the equation: $p = 20 + 5x - 3x^2$, where $x$ is the number of units demanded and $p$ is the price per unit (in Rs.), then the marginal revenue when 2 units are sold is:

The equation of the tangent line to the curve $y = x^2 - 2x + 5$ which is parallel to the line $4x - y + 1 = 0$ is

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

The function $f(x) = x + \frac{1}{x}$ has

If the maximum value of the function $f(x) = \frac{2\log_e x}{x}$, $x > 0$ occurs at $x = e$, then $e^3 f''(e)$ is equal to

For the function $f(x) = 2x^3 - 3x^2 - 12x + 5$, the difference of maximum and minimum value of $f(x)$ is

Interval in which the function $f$ given by $f(x) = \tan x - 4x$, $x \in (0, \frac{\pi}{2})$ is strictly decreasing is

The marginal cost (MC) and marginal revenue (MR) functions of a product are $MC = 20 + \frac{x}{20}$ and $MR = 30$ respectively. If the fixed cost is 200, then the maximum value of the profit is:

For the function, $f(x) = \frac{-3}{4}x^4 - 8x^3 - \frac{45}{2}x^2 - 350$, which of the following statements are correct?

(A) $x = -3$ and $x = -5$ are the only critical points of the given function.

(B) $x = -3$ is a point of local minimum.

(C) The local minimum value at $x = -3$ is 23.1.

(D) $x = -5$ is a point of local maximum.

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The real valued function $f(x) = x^{15} + 5x^9 + 10$ is increasing for___________.

If $y = ax^2 + bx$ has minima at $x = 2$ and the minimum value is -12, then which of the following are correct?

(A) $a = 3$

(B) $a = -3$

(C) $b = 12$

(D) $b = -12$

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The interval in which the function $f(x) = 2x^3 + 3x^2 - 12x + 1$ is strictly increasing, is