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

If the function $f(x)=\sin x+\cos x, 0 \le x \le 2\pi$ is increasing in the interval $(\alpha,\beta) \cup (\gamma,\delta)$, then $\alpha+\beta+\gamma+\delta$ is equal to:

The local maximum value of the function $f(x)=\dfrac{b^2}{a^4}(a^2x^2-x^4), a,b>0$ is equal to:

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If an edge of a variable cube is increasing at the rate of 0.8 cm/sec, at what rate its surface area is increasing when its edge is 10 cm?

The revenue function is given by $R(x)=150x-2x^2-x^3$, where $x$ is the number of units demanded, then the marginal revenue when 5 units are sold, is :

For the function $f(x) = 3x^4 + 4x^3 - 12x^2 + 12$, then

Match the LIST-I with LIST-II

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For the function $f(x) = |\sin 4x + 3|$, then which of the following statements are correct ?

A. Minimum value of $f(x)$ is 2

B. There is no minimum value

C. Maximum value of $f(x)$ is 4

D. Maximum value of $f(x)$ is 3

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Consider the function $f(x) = x^4 - 2x^2$, then which of the following are TRUE?

A. $f(x)$ is increasing on $(-1,0)$

B. $f(x)$ is decreasing on $(0,1)$

C. $f(x)$ is increasing on $(-\infty,-1)$

D. $f(x)$ is decreasing on $(1,\infty)$

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The volume of a cube is increasing at the rate 9 $cm^3/s$. How fast is the surface area increasing when the length of an edge is 10 cm ?

If a manufacturer's total cost 'C' is given by $C = \dfrac{x^2}{25} + 3x$, then the marginal cost when 7 units are produced will be:-

Which of the following statements are correct ?

A. The function $f(x) = a^x$ is increasing on $(-\infty,\infty)$ if $a>1$.

B. $f(x) = x^2 - 2x$ is decreasing in $(-\infty,1)$.

C. $f(x) = x(x-3)^2$ increases for $1 \le x \le 3$.

D. $f(x) = e^{-x}$ decreases in $[0,\infty)$.

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The maximum value of $f(x) = x^{50} - x^{20}$ in the interval $[0,1]$, is

If $f(x) = x^3 + px^2 + qx - 3$ has a local maxima at $x = 0$ and local minima at $x = 1$, then the values of $p$ and $q$ respectively, are:

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

The points in the interval $[0,2\pi]$, where the function $f(x) = \cos2x + 2\sin x\cos x$, attains its local minima, is/are:

A. $\pi/8$

B. $5\pi/8$

C. $13\pi/8$

D. $11\pi/8$

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A particle moves along a curve $3y^2 = 6x^3 + 8$ such that its y-coordinate is changing at the rate of 9 cm/sec at point (1,2). Then the rate of a change of x-coordinate at the same point is :

Let $C(x)$ and $R(x)$ be the cost function and the revenue function for $x$ units of a commodity such that marginal cost of producing 2 units is equal to marginal revenue of selling 1 unit. If $C(x) = ax^3 + bx^2 - 12x + 6$ and $R(x) = 100 + 6ax - bx^2$, then which one of the following is TRUE?

If the critical points (extreme values) of the function $f(x)$ given by $f(x) = -\dfrac{3}{4}x^4 - ax^3 - \dfrac{bx^2}{2} + 15$ are -3 and -5, then the value of $f''(-1)$ is :

For the function $f(x) = (x+1)^2(x-2)^2$, which of the following statements are TRUE?

(A) $f(x)$ is decreasing on $(-\infty,-1)$

(B) $f(x)$ is increasing on $\left(-1,\frac{1}{2}\right)$

(C) $f(x)$ is decreasing on $\left(\frac{1}{2},2\right)$

(D) $f(x)$ is increasing on $(2,\infty)$

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If the function $f(x)$ defined by $f(x) = x^3 - 3x + 100, (x \in \mathbb{R})$, (Where $\mathbb{R}$ is set of real numbers), then $f(x)$ has

Let the median of a variable equilateral triangle is increasing at the rate of $2\sqrt{3}$ cm/s. Then the rate at which its each side is increasing is:

The interval in which $f(x)=\frac{x}{\log x}$ is increasing, is:

If the local maximum value of the function $f(x)=x^3-2\lambda x^2+\lambda^2 x, \lambda>0$ is 108, Then the local minimum value of the function occurs at

The cost function of a product is given by $C(x)=\frac{1}{3}x^3-45x^2-900x+36$, where x is the number of units produced. In order to minimize the marginal cost, how many units of the product must be produced?

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

At x = 1, the function $f(x)=x^4-48x^2+\alpha x+120$ attains its maximum value in the interval [0, 2], then the value of $\alpha$ is:

The function $f:R\to R$ is defined by $f(x)=2x^3+5$, then which of the following statements are correct ?

(A) $f(x)$ has no local maximum value.

(B) $f(x)$ has no local minimum value.

(C) $f(x)$ has both local maximum and local minimum values.

(D) $f(x)$ has neither a local maximum value nor a local minimum value

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The interval for which the function $f(x)=\frac{x}{x^2+x+1}$ is an increasing function, is:

The function $f(x)=\log_e(\cos x)$ increases on which of the following intervals for $x\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ ?

An angle $\theta, 0<\theta<\frac{\pi}{2}$, which increases twice as fast as its sine, is