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

The value of the integral $\int_{-3}^{3} (x^3 - x) dx$ 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 x is real, the minimum value of $f(x) = x^2 - 8x + 20$ is :

If $x = 2at, y = at^2$, where 'a' is a constant, then $\frac{d^2 y}{dx^2}$ at $x = 2$ 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 :

If the total cost and total revenue of a company that produces and sells $x$ units of a particular product are $C(x) = 5x + 350$ and $R(x) = 50x - x^2$ respectively, then which of the following is/are the breakeven values:

(A) $x = 10$

(B) $x = 25$

(C) $x = 45$

(D) $x = 30$

Choose the correct answer from the options given below: