If the area of a circular sector of perimeter $60 \ m$ is to be maximized,then its radius must be......... $m$.

Two positive numbers $x$ and $y$ are such that $(x+y)=60$ and $x y^3$ is maximum. Then the numbers $x$ and $y$ are respectively:

Find the absolute maximum value and the absolute minimum value of the function given by $f(x) = 4x - \frac{1}{2}x^2$ for $x \in \left[-2, \frac{9}{2}\right]$.

$20 \ m$ of wire is available to fence a flowerbed in the form of a circular sector. If the flowerbed is to have maximum surface area,then the radius of the circle is (in $m$)

If the function $f(x) = \frac{x}{5} + \frac{5}{x}, (x \neq 0)$ attains its relative maximum value at $x = a$,then $\sqrt{a^2 + 2a - 6} = $

Observe the statements given below :
Assertion $(A)$ : $f(x)=x e^{-x}$ has the maximum at $x=1$
Reason $(R)$ : $f^{\prime}(1)=0$ and $f^{\prime \prime}(1) < 0$
Which of the following is correct?