Find the minimum value of $64 \sec x + 27 \csc x$ for $0 < x < \frac{\pi}{2}$.

At what value of $x$ is the function $f(x) = x(1 - x^2)$ maximum for $0 \leq x \leq 2$?

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

Let a function $f:[0,5] \rightarrow R$ be continuous. $f(1)=3$ and $F$ be defined as $F(x)=\int_{1}^{x} t^{2} g(t) dt$,where $g(t)=\int_{1}^{t} f(u) du$. Then for the function $F$,the point $x=1$ is

The maximum area (in sq. units) of a rectangle having its base on the $x-$axis and its other two vertices on the parabola $y = 12 - x^2$ such that the rectangle lies inside the parabola,is

The lower corner of a leaf in a book is folded over so as to just reach the inner edge of the page. The fraction of width folded over if the area of the folded part is minimum is