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

Let height and radius of a right circular cylinder are $h$ and $r$ respectively. If it is open at the top, having a given surface area and greatest volume, then

Sand is pouring from a pipe at the rate of 12 $cm^3$/sec. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4 cm ?

The interval in which $f(x)=-2x^3-9x^2-12x+1$ is increasing, is:

A steel plant is capable of producing 'x' tonnes per day of low-grade steel and 'y' tonnes per day of high-grade steel, where $y=\frac{32-5x}{10-x}$. If the fixed price of low-grade steel is half that of high-grade steel, then the quantity of low-grade steel that should be produced per day for maximum receipts is

Consider the function $f(x) = -2x^3 + 3x^2 + 36x - 15$, then which of the following are TRUE?

A. $f(x)$ increases on $(-2, 3)$

B. $f(x)$ decreases on $(-\infty, -2) \cup (3, \infty)$

C. $f(x)$ increases on $(-\infty, -2) \cup (3, \infty)$

D. $f(x)$ neither increases nor decreases on $(-2, 3)$

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Which of the following functions are strictly decreasing on $\left(0, \dfrac{\pi}{2}\right)$?

A. $\cos x$

B. $-\cot x$

C. $\cos 2x$

D. $\sin x$

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The point on curve $y^2 = 4x$ which is nearest to the point $(2, -8)$ is

The difference of absoulte maximum (largest) and absoulte minimum (smallest) values of the function $f(x) = 12x^{4/3} - 6x^{1/3}, x \in [-1, 1]$ is

The two positive numbers whose sum is 40 and the product of the cube of one number and the other number is maximum, are

The interval in which the function $f(x) = x^2 - 3x + 7$ is increasing, is

The points of local maxima and local minima of the $f(x) = x^3 - 6x^2 + 9x + 27$ are

The function $f(x) = 4x + \frac{1}{x}$ is

(A). Increasing in the interval $\left(-\infty, -\frac{1}{2}\right)$

(B). Increasing in the interval $\left(-\frac{1}{2}, 0\right)$

(C). Decreasing in the interval $\left(0, \frac{1}{2}\right)$

(D). Decreasing in the interval $\left(\frac{1}{2}, \infty\right)$

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Given a function $f(x) = ax + \frac{b}{x}$; $a > 0, b > 0$, then

Match List-I with List-II

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The function $f(x) = \sin x + \cos x$; $0 \leq x \leq 2\pi$ is

(A) Increasing in $\left(0, \frac{\pi}{4}\right)$

(B) Decreasing in $(0, \pi)$

(C) Increasing in $\left(\frac{5\pi}{4}, 2\pi\right)$

(D) Decreasing in $\left(\frac{\pi}{4}, \frac{5\pi}{4}\right)$

(E) Neither increasing Nor decreasing in $\left(0, \frac{\pi}{4}\right)$

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A balloon which always remains spherical is being inflated by pumping in 900 cubic cm of gas per second. The rate at which the radius of the balloon increases when the radius is 15 cm is

The points on the ellipse $4x^2 + y^2 = 4$ that are farthest from the point (1,0) are;

Match List-I with List-II

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The maximum and the minimum value of the function $f(x) = -||x+3|-5| + 2$ are :

Function $f(x) = x + \frac{1}{x}, x \neq 0$ is increasing in the interval

The absolute maximum value of the function $f(x) = \frac{x}{2} + \frac{2}{x}$ in the interval $[-3, 3]$, is

Let $f(x) = x^3 - 3ax$, where $a > 0$, then the values for which $f(x)$ is decreasing, are

If the sum of the length of the hypotenues and a side of a right angled triangle is L cm then the angle between hypotenuse and other side of the triangle for which the area of the triangle is maximum, is:

The demand function for a certain product is represented by $p = 35 + 5x - 3x^2$, where $x$ is the number of units demanded and $p$ is the price per unit. The marginal revenue when 2 units are sold is

The interval in which the function $f(x) = -x^2 - 2x + 18$ is decreasing is :

The number of points of the local maximum value of the function $f(x) = 3x^4 + 4x^3 - 12x^2 + 2$ is,

The maximum area of a rectangle inscribed in a circle of radius $3\sqrt{2}$ cm, is:

For the function $f(x) = \frac{x^4}{4} - 2x^3 + \frac{11}{2}x^2 - 6x$, which of the following statements are TRUE?

(A) Critical points of $f(x)$ are $x = 1, 2$ and $3$.

(B) $f(x)$ is increasing over $(1,2)$.

(C) $f(x)$ is decreasing over $(2,3)$.

(D) $f(x)$ is increasing over $(3,\infty)$.

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

The function $f(x) = 3x + \cos 3x$ is (where R is set of real numbers)

For $f(x) = x^3 - 6x^2 + 9x + 15$, which of the following statements are TRUE?

(A) $f'(x) = 3x^2 - 12x + 15$.

(B) The critical points are 3 and 1.

(C) $x = 1$ is a point of local maxima.

(D) Local minimum value is 19.

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