The abscissae of the points of the curve $y = x(x - 2)(x - 4)$ where the tangents are parallel to the $x$-axis are obtained as:

Find the maximum and the minimum values,if any,of the function $f$ given by $f(x) = x^{2}, x \in R$.

Two positive numbers $x$ and $y$ are such that $(x+y)=60$ and $x y^3$ is maximum. Then the numbers $x$ and $y$ are respectively:

The sum of all the local minimum values of the twice differentiable function $f: R \rightarrow R$ defined by $f(x)=x^{3}-3 x^{2}-\frac{3 f^{\prime \prime}(2)}{2} x+f^{\prime \prime}(1)$ is:

The maximum value of $f(x) = (x + 1)^{\frac{1}{3}} - (x - 1)^{\frac{1}{3}}$ for $x \in [0, 1]$ is ....

Let $a, b \in R$ be such that the function $f(x) = \ln|x| + bx^2 + ax, x \neq 0$ has extreme values at $x = -1$ and $x = 2$.
Statement-$1$: $f$ has a local maximum at $x = -1$ and $x = 2$.
Statement-$2$: $a = \frac{1}{2}$ and $b = -\frac{1}{4}$.