Let $f(x)=(x-2)^{17}(x+5)^{24}$. Then

$A$ window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is $10 \, m$. Find the dimensions of the window to admit maximum light through the whole opening.

The minimum value of the slope of the tangent to the curve $y=x^3-3x^2+2x+93$ is

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

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 $f'(a) \neq 0$.

Let $f(x)=3 \sqrt{x-2}+\sqrt{4-x}$ be a real-valued function. If $\alpha$ and $\beta$ are respectively the minimum and the maximum values of $f$,then $\alpha^2+2 \beta^2$ is equal to