The minimum value of $f(a) = (2a^2 - 3) + 3(3 - a) + 4$ is

The function $f(x) = x^5 - 5x^4 + 5x^3 - 10$ has a local maximum at $x =$

Let $f(x) = x \sin \pi x$,$x > 0$. Then for all natural numbers $n$,$f^{\prime}(x)$ vanishes at
$(A)$ a unique point in the interval $\left(n, n+\frac{1}{2}\right)$
$(B)$ a unique point in the interval $\left(n+\frac{1}{2}, n+1\right)$
$(C)$ a unique point in the interval $(n, n+1)$
$(D)$ two points in the interval $(n, n+1)$

Consider the region $R = \{(x, y) : x \leq y \leq 9 - \frac{11}{3} x^2, x \geq 0\}$. The area of the largest rectangle with sides parallel to the coordinate axes and inscribed in $R$ is:

$A$ 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 $y = \alpha \log x + \beta x^3 - x$ has extreme values at $x = -1$ and $x = 1$,then $\alpha$ and $\beta$ are respectively