If $x$ and $y$ are two positive integers such that $x+y=24$ and $x^3 y^5$ is maximum, then $x^2+y^2=$

The maximum volume (in cubic units) of the cylinder which can be inscribed in a sphere of radius $12$ units is

Let $f:[-1,1] \rightarrow R$ be defined as $f(x)=ax^{2}+bx+c$ for all $x \in[-1,1],$ where $a, b, c \in R$ such that $f(-1)=2, f^{\prime}(-1)=1$ and for $x \in(-1,1)$ the maximum value of $f^{\prime\prime}(x)$ is $\frac{1}{2}.$ If $f(x) \leq \alpha$ for all $x \in[-1,1],$ then the least value of $\alpha$ is equal to:

The maximum volume (in $m^3$) of the right circular cone having slant height $3 \, m$ is

Let $f(x)$ be a polynomial of degree $5$ such that $x=\pm 1$ are its critical points. If $\mathop {\lim }\limits_{x \to 0} \left(2+\frac{f(x)}{x^{3}}\right)=4,$ then which one of the following is not true?

The function $f(x) = x^2 \log x$ in the interval $(1, e)$ has