If $y=a \log x+b x^2+x$ has its extreme value at $x=-1$ and $x=2$,then the value of $a+b$ is

The function $f(x) = 2x^3 - 9ax^2 + 12a^2x + 1$ where $a > 0$ attains its local maximum and local minimum at $p$ and $q$ respectively. If $p^2 = q$, then $a =$

$A$ sector is removed from a metallic disc and the remaining region is bent into the shape of a circular conical funnel with volume $2 \sqrt{3} \pi$. The least possible diameter of the disc is

Let $x_0$ be the point of local minima of $f(x) = \overline{a} \cdot (\overline{b} \times \overline{c})$ where $\overline{a} = x \hat{i} - 2 \hat{j} + 3 \hat{k}$,$\overline{b} = -2 \hat{i} + x \hat{j} - \hat{k}$,and $\overline{c} = 7 \hat{i} - 2 \hat{j} + x \hat{k}$. Then the value of $\overline{a} \cdot \overline{b}$ at $x = x_0$ is:

Find the maximum value of $2x^{3}-24x+107$ in the interval $[1,3]$. Also,find the maximum value of the same function in the interval $[-3,-1]$.

$x$ and $y$ are two variables such that $x > 0$ and $xy = 1$. Then the minimum value of $x + y$ is