If the function $f(x) = 2x^3 - 9ax^2 + 12a^2x + 1$,where $a > 0$,attains its maximum and minimum at $p$ and $q$ respectively such that $p^2 = q$,then $a$ equals

Find two numbers whose sum is $24$ and whose product is as large as possible.

The maximum slope of the curve $y = \frac{1}{2} x^{4} - 5 x^{3} + 18 x^{2} - 19 x$ occurs at the point

Statement-$I$: The sequence $a_n = \frac{n^2}{n^3 + 200}, n \in N$ has its $7^{th}$ term as the largest term.
Statement-$II$: The function $f(x) = \frac{x^2}{x^3 + 200}$ attains a local maximum at $x = 7$.

Let $f(x) = x^3 + ax^2 + bx + c$ where $a, b, c$ are real numbers. If $f(x)$ has a local minimum at $x = 1$ and a local maximum at $x = -\frac{1}{3}$ and $f(2) = 0$,then $\int_{-1}^1 f(x) dx$ equals

$P(x) = x^4 + ax^3 + bx^2 + cx + d$ is such that $x = 0$ is the only real root of $P'(x) = 0$. If $P(-1) < P(1)$,then in the interval $[-1, 1]$: