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

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

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

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

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:

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

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