Question

What is the maximum of `f(x)= sqrt((sinx + cosx)^2)`?

Answer 1

The maximum will occur at `y = sqrt(2)`

Explanation:

Letting the function be `f(x)`, we can see that

`f(x) = sqrt((sinx + cosx)^2)`

`f(x) = sqrt(sin^2x + cos^2x+ 2sinxcosx)`

`f(x) = sqrt(1 + 2sinxcosx)`

`f(x) = sqrt(sin(2x) + 1)`

The maximum of `sin(2x) + 1` will be `y =2`. Therefore the maximum of `sqrt(sin(2x) + 1)` will be `sqrt(2)`.

A graphical verification will confirm:

Hopefully this helps!