Question

Hello everyone, does anybody know:

how to prove that for any number of X is the real inequality.


`Sin^(8)x + Cos^(8)x \geq (1)/(8)`

My calculations below, but I didn’t have the idea what shall I do next

Answer 1

Let
`y=\cos^2x`. Then


`\cos^8x+\sin^8x=\cos^8x+(1-\cos^2x)^4=y^4+(1-y)^4`

If calculus methods are okay, consider the function
`f(y)=y^4+(1-y)^4`. Then
`f` has a critical point where
`f'` vanishes, i.e.



`f'(y)=4y^3-4(1-y)^3=0\implies y^3=(1-y)^3\implies y=\frac12`

The second derivative
`f''` is


`f''(y)=12y^2+12(1-y)^2`

and at this critical point, we have a value of


`f''\left(\frac12\right)=6>0`. The derivative test for extrema indicates this is the site of a minimum, and we get a minimum value of


`f\left(\frac12\right)=\frac18`

as desired.