What is the minimum value of:
$$(\sin x + \cos x + \csc 2x )^3$$
let us consider that $0<x<\pi/2$
Use of differentiation is not allowed.!
Now we can use am gm to get the minimum value as 13.5...!!But the problem is that for this to holds $\sin x = \cos x = \csc(2x)$ which is not possible..
As $$lim_{x\to\frac\pi4^+}\sec 2x=-\infty$$ $\sin x+\cos x$ is bounded and $t\mapsto t^3$ is increasing, there is no minimum in $(0,\pi/2)$.
But in the interval $[0,\pi/4)$: $$\sin x+\cos x=\sqrt 2 \sin(x+\pi/4)$$ is increasing and $\sec 2x\ $ is also increasing, so their sum composed with the cube is increasing. Because this, the minimum is reached at $x=0$: $$(\sin 0+\cos x+\sec 0)^3=2^3=8.$$