If from a wire of length $36 \ m$ a rectangle of greatest area is made,then its two adjacent sides in metres are

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

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 $f'(a) \neq 0$.

Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius $R$ is $\frac{2 R}{\sqrt{3}} .$ Also find the maximum volume.

For all real values of $x,$ the minimum value of $\frac{1-x+x^{2}}{1+x+x^{2}}$ is

The number of values of $x$ where the function $f(x) = \cos x + \cos (\sqrt{2} x)$ attains its maximum is