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If $y = x^3 \log x$, then $\frac{d^2 y}{dx^2}$ is equal to :

The value of the integral $\int e^x \left(\frac{1}{x} - \frac{1}{x^2}\right) dx$ is :

Area of the region bounded by the curve $x^2 = 4y$, $x$ - axis and $x = 3$ is

The value of the integral

$I = \int_{-1}^{1} (x + x^3 + x^5) dx$ is :

If $(x+ 1) e^y = 1$ , then :

Match List - I with list- II

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The value of a for which the function $f(x) = a^x$ is increasing on R are given by :

Match List - I with list- II

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Any function f(x) is an increasing function in [a,b] if :

(A) $x_1, x_2 \in [a, b], f(x_1) \geq f(x_2)$ if $x_1 < x_2$

(B) $x_1, x_2 \in [a, b], f(x_1) \geq f(x_2)$ if $x_1 > x_2$

(C) (A) $x_1, x_2 \in [a, b], f(x_1) \leq f(x_2)$ if $x_1 < x_2$

(D) (A) $x_1, x_2 \in [a, b], f(x_1) < f(x_2)$ if $x_1 > x_2$

If the cost function and the profit function for a company is given by $C = 10 - 0.3x^2$ and $P = 0.3x^2 + 2x - 10$ respectively , where X represent units of output, then the revenue function is given by :

A monopolist's Demand function is $x = 70 - \frac{P}{2}$, the revenue at $x = 5$ will be :

If $y = \log_3(\log_3 x)$, then $\frac{dy}{dx}$

If $t = e^{2x}$ and $\log_e t^2$, then $\frac{d^2y}{dx^2}$ is :

If $x = at^2$ and $y = a^3 t^3$, then $\frac{d^2 y}{dx^2}$ will be :

The minimum value of $f(x) = 4x^3 - 48x + 105$ in the interval [1,3] is :

For a manufacturer, total cost function is given by $C = \frac{x^2}{25} + 2x$. Which of the following are correct ?

A. 2.6 is the marginal cost when 5 units are produced.

B. $\frac{2x}{25} + 2$ is the marginal cost function.

C. $\frac{x}{25} + 2$ is the average cost function.

D. 2.4 is the marginal cost when 5 units are produced.

E. $\frac{x}{25} + 1$ is the average cost function.

Choose the correct answer from the options given below:

Two positive numbers $x$ and $y$ such that $x + y = 60$ and $xy^3$ is maximum are :

The total cost of producing $x$ generators is given by

TC = $x^3 - 60x^2 + 1500x + 2000$.

The Marginal Cost (MC), when $x = 10$ units is:

If $y = \frac{\log x}{x^2}$, then $\frac{d^2y}{dx^2}$ is equal to

If $x^{2/3} + y^{2/3} = a^{2/3}$, then $\frac{dy}{dx}$ is equal to :

Match List - I with List - II.

Choose the correct answer from the options given below :

Let $f : R \to R$ be a function defined as $f(x) = 2x^3 - 21x^2 + 36x - 20$, then :

(A) maximum value of $f(x)$ is $-3$

(B) minimum value of $f(x)$ is $-128$

(C) maximum value exists at $x = 6$

(D) minimum value exists at $x = 1$

Choose the correct answer from the options given below :

If $y = 3e^{2x} + 2e^{3x}$, then which one of the following is true ?

Match List - I with List - II.

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If $x = 2t^2 + 3, y = 3t^2 + 6t + 5$, then the value of $\frac{d^2y}{dx^2}$ is :

Let $f : R \to R$ be defined such that $f(x) = 16x^2 - 16x + 12$

(A) Maximum value of $f(x)$ is 8

(B) Minimum value of $f(x)$ is 8

(C) Minimum value of $f(x)$ is 16

(D) No maximum value of $f(x)$

Choose the correct answer from the options given below :

If $C(x) = ax^2 - bx - c$ represents the total cost function then the slope of the tangent to the marginal cost curve at the point $(x, y)$ is :

The demand function of a monopolist is given by $p = 1500 - 2x - x^2$, then value of marginal revenue when $x = 20$ is :

Match List - I with List - II.

Choose the correct answer from the options given below :

If $x^3 + y^3 = xy$, then $\frac{dy}{dx}$ is equal to :