The maximum value of $xy$ subject to $x + y = 8$ is

Divide $20$ into two parts such that the product of one part and the cube of the other is maximum. The two parts are

The sum of the absolute minimum and the absolute maximum values of the function $f(x) = |3x - x^2 + 2| - x$ in the interval $[-1, 2]$ is

Let $a > 0$. If the function $f(x) = 6x^3 - 45ax^2 + 108a^2x + 1$ attains its local maximum and minimum values at the points $x_1$ and $x_2$ respectively such that $x_1x_2 = 54$,then $a + x_1 + x_2$ is equal to:

The shortest distance between the point $\left( \frac{3}{2}, 0 \right)$ and the curve $y = \sqrt{x}, (x > 0)$ is:

Prove that the function $h(x) = x^{3} + x^{2} + x + 1$ does not have any local maxima or local minima.