Find all the points of local maxima and local minima of the function $f$ given by $f(x) = 2x^3 - 6x^2 + 6x + 5$.

$A$ rectangular sheet of fixed perimeter with sides having their lengths in the ratio $8:15$ is converted into an open rectangular box by folding after removing squares of equal area from all four corners. If the total area of removed squares is $100$,the resulting box has maximum volume. The lengths of the sides of the rectangular sheet are:
$(A)$ $24$
$(B)$ $32$
$(C)$ $45$
$(D)$ $60$

The difference between the greatest and the least values of the function $f(x) = x(\ln x - 2)$ on the interval $[1, e^2]$ is:

The function $f(x)=2|x|+|x+2|-||x+2|-2|x||$ has a local minimum or a local maximum at $x=$

The sum of the perimeter of a circle and a square is $k$,where $k$ is a constant. Prove that the sum of their areas is least when the side of the square is double the radius of the circle.

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