If $x + y = 16$ and $x^2 + y^2$ is minimum,then the values of $x$ and $y$ are

The figure shown below is the graph of the derivative of some function $y=f(x)$. Then,

The sum of two numbers is $20$. If the product of the square of one number and the cube of the other is maximum, then the numbers are:

The number of turning points of the curve $f(x) = 2 \cos x - \sin 2x$ in the interval $[-\pi, \pi]$ is

If $m$ and $n$ respectively are the number of local maximum and local minimum points of the function $f(x) = \int_{0}^{x^{2}} \frac{t^{2}-5t+4}{2+e^{t}} dt$,then the ordered pair $(m, n)$ is equal to

Let a rectangle $ABCD$ of sides $2$ and $4$ be inscribed in another rectangle $PQRS$ such that the vertices of the rectangle $ABCD$ lie on the sides of the rectangle $PQRS$. Let $a$ and $b$ be the sides of the rectangle $PQRS$ when its area is maximum. Then $(a+b)^2$ is equal to :