$A$ wire of length $64 \text{ m}$ is to be bent to form a rectangle such that its area is maximum. What is the maximum area (in $\text{ m}^2$)?

If $f(x) = 7e^{\sin^2 x} - e^{\cos^2 x} + 2$,then $\sqrt{7f_{\min} + f_{\max}}$ is equal to

The set of value$(s)$ of $a$ for which the function $f(x) = \frac{a x^3}{3} + (a + 2) x^2 + (a - 1) x + 2$ possesses a negative point of inflection.

If the function $f(x) = (\frac{1}{x})^{2x}$ for $x > 0$ attains the maximum value at $x = \frac{1}{e}$,then:

The range of $a \in R$ for which the function $f(x)=(4 a-3)\left(x+\log _{e} 5\right)+2(a-7) \cot \left(\frac{x}{2}\right) \sin ^{2}\left(\frac{x}{2}\right)$ where $x \neq 2 n \pi, n \in N$,has critical points,is

Let $f : R \rightarrow R$ be a function defined by $f(x) = ||x+2|-2|x||$. If $m$ is the number of points of local minima and $n$ is the number of points of local maxima of $f$,then $m+n$ is