If $6x - x^2 + 12$ attains its extreme value $\beta$ at $x = \alpha$, then $\beta =$

$A$ 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 $\text{ m}^2$)?

Suppose that $f$ is a polynomial of degree $3$ and that $f''(x) \neq 0$ at any of the stationary points. Then

Let the sum of the maximum and the minimum values of the function $f(x) = \frac{2x^2 - 3x + 8}{2x^2 + 3x + 8}$ be $\frac{m}{n}$,where $\gcd(m, n) = 1$. Then $m + n$ is equal to :

Find the local maximum and local minimum values for the function given by $g(x) = \frac{x}{2} + \frac{2}{x}, \quad x > 0$.

The maximum value of the function $f(x) = -|x+1| + 3, x \in R$ is . . . . . . .