An enemy Apache helicopter is flying along the curve given by $y = x^{2} + 7$. $A$ soldier,placed at $(3, 7)$,wants to shoot down the helicopter when it is nearest to him. Find the nearest distance.

Find the positive value of $a$ for which the equality $2 \alpha + \beta = 8$ holds,where $\alpha$ and $\beta$ are the points of maximum and minimum,respectively,of the function $f(x) = 2 x^3 - 9 a x^2 + 12 a^2 x + 1$.

Find the local maximum and local minimum values for the function given by $f(x) = x^{2}$.

Let $S=(-1, \infty)$ and $f: S \rightarrow R$ be defined as $f(x)=\int_{-1}^x (e^t-1)^{11}(2t-1)^5(t-2)^7(t-3)^{12}(2t-10)^{61} dt$. Let $p$ be the sum of the squares of the values of $x$ where $f(x)$ attains local maxima on $S$,and $q$ be the sum of the values of $x$ where $f(x)$ attains local minima on $S$. Then,the value of $p^2+2q$ is

Let $f: R \rightarrow R$ be given by $f(x) = |x^{2} - 1|$, $x \in R$. Then:

If for a function $f(x)$,$f'(a) = 0$ and $f''(a) = 0$,then $x = a$ is: