The number of points at which the function $f(x) = \int\limits_0^x {{e^{t - 3}}} \left( {{t^2} + 2} \right)\left( {t - 3} \right){\left( {t + 4} \right)^2}dt$ has a local minimum is:

For a positive real number $x$,when its reciprocal is added to it,the sum attains a minimum value. What is the value of $x$?

If the function $f(x)=2 x^{3}-9 a x^{2}+12 a^{2} x+1$ attains its maximum and minimum at $p$ and $q$ respectively such that $p^{2}=q$,then $a$ equals

If $m$ and $M$ respectively denote the minimum and maximum of $f(x)=(x-1)^2+3$ for $x \in [-3, 1]$,then the ordered pair $(m, M)$ is equal to

If the function $f(x) = a \sin(x) + \frac{1}{3} \sin(3x)$ attains its maximum value at $x = \frac{\pi}{3}$,then $a$ equals:

In an isosceles trapezium,the length of one of the parallel sides and the lengths of the non-parallel sides are all equal to $30$. In order to maximize the area of the trapezium,the smallest angle should be: