The maximum slope of the curve $y = \frac{1}{2} x^{4} - 5 x^{3} + 18 x^{2} - 19 x$ occurs at the point

The sufficient conditions for the function $f:R \to R$ to have a local maximum at $x = a$ are:

The function $S(x) = \int\limits_0^x {\sin \left( {\frac{{\pi {t^2}}}{2}} \right)\,dt} $ has two critical points in the interval $[1, 2.4]$. One of the critical points is a local minimum and the other is a local maximum. The local minimum occurs at $x =$

The maximum volume of the right circular cone with slant height $6$ units 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 $f'(0) = 0$.

If ${a^2}{x^4} + {b^2}{y^4} = {c^6}$,then the maximum value of $xy$ is