If $f(x) = x^4 + \lambda x^3 + x^2$ $(\lambda \in R)$ has a local maximum at $x = \frac{1}{2}$,then the absolute minimum value of $f(x)$ is:

Let $f(x)$ be a polynomial of degree $3$ such that $f(-1)=10$,$f(1)=-6$,$f(x)$ has a critical point at $x=-1$,and $f^{\prime}(x)$ has a critical point at $x=1$. Then $f(x)$ has a local minima at $x=$

If $x - 2y = 4,$ the minimum value of $xy$ is

Let the radius and height of a right circular cylinder be related as $r^2 + h = 6$. If the volume of the cylinder is maximum,then the value of $\frac{r}{h}$ is:

Let $p(x)$ be a real polynomial of least degree which has a local maximum at $x=1$ and a local minimum at $x=3$. If $p(1)=6$ and $p(3)=2$, then $p^{\prime}(0)$ is equal to

Find two positive numbers whose sum is $15$ and the sum of whose squares is minimum.