The function $f(x) = \frac{x}{2} + \frac{2}{x}$ has a local minimum at $x = $ ........

The function $f(x) = x\sqrt{1 - x^2}$ for $x > 0$ has:

Find the maximum and the minimum values,if any,of the function given by $f(x) = x, x \in (0, 1)$.

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 $f'(a) \neq 0$.

The sum of the terms of an infinitely decreasing geometric progression is equal to the greatest value of the function $f(x) = x^3 + 3x - 9$ on the interval $[-2, 3]$. If the difference between the first and the second term of the progression is equal to $f'(0)$,then the common ratio of the $G.P.$ is

Find the absolute maximum and minimum values of the function $f$ given by $f(x) = \cos^{2} x + \sin x$,where $x \in [0, \pi]$.