In a regular triangular prism,the distance from the centre of one base to one of the vertices of the other base is $l$. Find the altitude of the prism for which the volume is greatest.

Let $f(x) = x^3 e^{-3x}, x > 0$. Then the maximum value of $f(x)$ is

If from a wire of length $36 \ m$ a rectangle of greatest area is made,then its two adjacent sides in metres are

Let $f(x) = x^3 + px + 1$ and consider the following three statements:
$(i)$ For $p \geqslant 0$,$f(x) = 0$ has one negative root and $f(x)$ is monotonic.
$(ii)$ For $-1 < p < 0$,$f(x) = 0$ has one negative root and $f(x)$ is non-monotonic.
$(iii)$ For $p < -3/\sqrt[3]{4}$,$f(x) = 0$ has three real and distinct roots.
Which of the following is correct?

If $f(x) = 7e^{\sin^2 x} - 7e^{\cos^2 x} + 2$,then $\sqrt{7f_{\min} + f_{\max}}$ is equal to:

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