Question

If x = sin¹⁴ A + cos^20 A, then
Prove that 0 < x ≤ 1. For all A belongs to Real Number.

Answer 1

the function sin^14(x)+cos^20(x) will be just like a oscillating function b/w its extremes as it is a periodic function .
find the derivative of it , you will get
14sin^13(x)cos(x) - 20sin(x) cos^19(x),
it is zero when x is either nπ or (2n+1)π/2 , n€N
the function value at both nπ and (2n+1)π/2 is 1
and also it is very clear that it can have only positive values,
so we have
$0 < function \leqslant 1$