If the cost function $C(x)$ of producing $x$ units of a commodity is given as $C(x) = x^3 - 60x^2 + 13x + 50$, then the level of output for which the marginal cost is minimum is

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 :

Match the LIST-I with LIST-II

Choose the correct answer from the options given below:

$\int \frac{dx}{x^{n+1} - x}\,dx$

The degree of the differential equation $\left(1 - \left(\frac{dy}{dx}\right)^2\right)^{3/2} = k \frac{d^2 y}{dx^2}$ is :

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 $v = \frac{4}{3} \pi r^3$, at what rate is cubic / unit sec is increasing when $r = 10$, and $\frac{dr}{dt} = 0.01$ ?