If for a function $f(x)$,$f'(a) = 0$ and $f''(a) = 0$,then $x = a$ is:

Let $f : (-\infty, \infty) \to (-\infty, \infty)$ be defined by $f(x) = x^3 + 1$.
Statement-$1$: The function has a local extremum at $x = 0$.
Statement-$2$: The function $f(x)$ is continuous and differentiable on $(-\infty, \infty)$ and $f'(0) = 0$.

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 $P(x)$ be a real polynomial of degree $3$ which vanishes at $x = -3$. Let $P(x)$ have local minima at $x = 1$,local maxima at $x = -1$,and $\int_{-1}^{1} P(x) dx = 18$. Then the sum of all the coefficients of the polynomial $P(x)$ is equal to ....... .

If local maximum of $f(x) = \frac{ax + b}{(x - 1)(x - 4)}$ exists at $(2, -1)$, then $a + b =$

The maximum value of $xy$ when $x+2y=8$ is