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

If $x$ is real, then the minimum value of $x^2 - 8x + 25$ is

The function $f(x) = x^3 + 2x^2 - 1$ is

(A) increasing in $\left(\dfrac{-4}{3}, 0\right)$

(B) decreasing in $\left(\dfrac{-4}{3}, 0\right)$

(C) increasing in $\left(-\infty, \dfrac{-4}{3}\right)$

(D) increasing in $(0, \infty)$

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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

The maximum value of $f(x)=\dfrac{1}{3(x^2-x+1)}$ is :

For the function $f(x)=ax^3-9ax^2+9x+3, x\in R$, then which of the following statements are True?

(A) $f'(x)=3[ax^2-6ax+3]$

(B) It is increasing for $a\in\left[0,\dfrac{1}{3}\right]\ \forall x\in R$

(C) It is increasing for $a\in(-\infty,0)\ \forall x\in R$

(D) It is increasing for $\forall a\in R$

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The total revenue received from the sale of $x$ units of a product is given by $R(x)=24x+5x^2+3$. The marginal revenue when $x=10$ is,

A ring is dropped into a quiet pond and waves move in circles at a speed of 2 cm/second. At the instant, when the radius of the circular wave is 12 cm, how fast is the enclosed area increasing?

A function $f(x)$ is defined as:

$f(x) = x^3-6x^2+9x+15$

For the function $f(x)$, which of the following statements are correct ?

(A) The critical points are $x=1$ and $x=3$.

(B) $x=1$ is a point of local minimum.

(C) The local maximum value is 19.

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

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Which of the following functions has a local minima at $x = 0$?

(A) $f(x) = x^3$

(B) $f(x) = |x|$

(C) $f(x) = x^2$

(D) $f(x) = x^{-2}$

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Function $f(x) = x^x, x > 0$ decreases on the interval

Let $f(x) = x^3 - 6x^2 + 9x - 8$ be a function, then which of the following statements are TRUE?

(A) $f'(x) = 3(x - 1)(x - 3)$

(B) The critical points of the function are $x = 1$ and $x = 3$

(C) $x = 1$ is the point of local minimum

(D) The local maximum value is $-4$

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The sides of an equilateral triangle are increasing at the rate of 5 cm/sec. The rate at which the area increases when the side is 20 cm, is

Let $f(x) = \log_e(\sin x), x \in (0, \pi)$, then which of the following statements is/are TRUE?

(A) $f(x)$ is increasing on $(0, \pi/2)$

(B) $f(x)$ is decreasing on $(\pi/2, \pi)$

(C) $f(x)$ is increasing on $(0, \pi)$

(D) $f(x)$ is decreasing on $(0, \pi)$

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The length of a rectangle is decreasing at the rate of 4 cm/minute and the width is increasing at the rate of 3 cm/minute, then the rate of change of the perimeter is

The two positive numbers whose sum is 16 and the sum of whose squares is minimum then the positive numbers are:

The demand for a certain product is represented by the function

$p = 150 + 10x - x^2$ (in Rs.)

where $x$ is the number of units demanded and $p$ is the price per unit, then the value of marginal revenue, when 10 units are sold is

The function $f(x) = 4x^3 - 7x^2$ has point(s) of local minima at

For the function $f(x) = x^x, x > 0$, which of the following are TRUE?

(A) $f'(x) = x^x(1 + \log x)$

(B) $x = e$ is the critical point

(C) $f$ is increasing in $(\frac{1}{e}, \infty)$

(D) $f$ is increasing in $(0, \infty)$

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The function $f(x) = 4 - 3x + 3x^2 - x^3$ is (Here $\mathbb{R}$ is set of real numbers)

Let $f(x) = x^2 + \frac{250}{x}$ be any function defined on $\mathbb{R} - \{0\}$, where $\mathbb{R}$ is the set of real numbers. Then which of the following are TRUE?

(A) $f'(x) = 2x + \frac{250}{x^2}$

(B) $x = 5$ in the only critical point of $f(x)$

(C) minimum value of $f(x)$ is 75

(D) maximum value of $f(x)$ is 50.

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The edge of a cube is increasing at a rate of 7cm/s. The rate of change of area of the cube when edge of the cube is 3cm is:

Let f be a function defined by $f(x) = 2x^3 - 3x^2 - 36x + 2$, then which of the following are correct?

(A) The critical points of f(x) are -2 and 3.

(B) The function f(x) increases in the interval $(3, \infty)$

(C) The function f(x) decreases in the interval (-2,3)

(D) The function f(x) increases in the interval (-2,3)

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If $x$ is real, the minimum value of $x^2 - 8x + 20$ is

For the function $f(x) = -2x^3 + 3x^2 + 36x - 10$, which of the following is/are true?

(A) $f$ is increasing in $(-\infty, -2)$

(B) $f$ is increasing in $(-2, 3)$

(C) $f$ is decreasing in $(-\infty, -2)$

(D) $f$ is decreasing in $(3, \infty)$

Choose the correct answer from the options given below:

Consider a closed cylinder of radius $r$ with a fixed surface area. The volume of the cylinder is maximum when its height is

The function $f(x) = 2\log_e(x-2) - x^2 + 4x + 1, (x > 2)$ is increasing on the interval:

If $x = -1$ and $x = -2$ are the extreme points of $f(x) = \alpha\log|x| + \beta x^2 + x$ then

If maximum value of $f(x) = 2x^3 + 3x^2 - 6ax + 10$ occurs at $x = -3$, then the value of $\alpha$ is ____

The function, $f(x) = x - \frac{1}{x}$ is

For $x > 0$, the minimum value of $\frac{x}{\log_e x}$ is