The minimum distance of a point on the curve $y=x^2-4$ from the origin is

Find two positive numbers $x$ and $y$ such that their sum is $35$ and the product $x^{2} y^{5}$ is a maximum.

Find the absolute maximum value and the absolute minimum value of the function given by $f(x) = (x - 1)^{2} + 3, x \in [-3, 1]$.

For every twice differentiable function $f : R \rightarrow [-2, 2]$ with $(f(0))^2 + (f'(0))^2 = 85$,which of the following statement$(s)$ is (are) $TRUE$?
$(A)$ There exist $r, s \in R$,where $r < s$,such that $f$ is one-one on the open interval $(r, s)$.
$(B)$ There exists $x_0 \in (-4, 0)$ such that $|f'(x_0)| \leq 1$.
$(C)$ $\lim_{x \rightarrow \infty} f(x) = 1$.
$(D)$ There exists $a \in (-4, 4)$ such that $f(a) + f''(a) = 0$ and $f'(a) \neq 0$.

If $f(x) = 1 + 2x^2 + 2^2 x^4 + \dots + 2^{10} x^{20}$,then $f(x)$ has:

The maximum area of the rectangle that can be inscribed in a circle of radius $r$ is: