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 function $f(x) = 2x^3 - 3x^2 - 36x + 7$ is :

A. Increasing in $(-\infty,-2)$

B. Decreasing in $(-2,3)$

C. Neither increasing nor decreasing in $(-2,3)$

D. Increasing in $(3,\infty)$

Choose the correct answer from the options given below:

If $f(x) = a \log_e|x| + bx^2 + x$ has critical points at $x = -2$ and $x = 1$, then

The sides of an equilateral triangle are increasing at the rate of 2 cm/sec. The rate at which the area increases when the side is 10 cm, is

Water is running into a conical vessel, 18cm deep and 6cm in radius, at the rate of 0.1 $cm^3$/sec. If the water is 4cm deep, then the water level rising at the rate

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.

Choose the correct answer from the options given below:

The total cost $c(x)$ associated with the production of $x$ units of an item is given by $c(x) = 0.001x^3 + 0.06x^2 + 20x + 500$. The marginal cost when 10 units are produced is: