For the curve $y = x e^x$,the point:

Find the absolute maximum value and the absolute minimum value of the function given by $f(x) = 4x - \frac{1}{2}x^2$ for $x \in \left[-2, \frac{9}{2}\right]$.

At what value of $x$ does the function $f(x) = x^x$ $(x > 0)$ attain its minimum value?

If $ab = 2a + 3b$ where $a > 0$ and $b > 0$,then the minimum value of $ab$ is:

$A$ triangular park is enclosed on two sides by a fence and on the third side by a straight river bank. The two sides having fence are of same length $x$. The maximum area enclosed by the park is

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?