Question

f(x) is concave down on which interval? For f(x) = sin^2 (x/5)

how about a global minimum and local maximum (which is not a global maximum) for this function?

which region is it increasing?

Answer 1

Take derivative, look at 0s. Positive slope before 0 and negative after 0 means high point, and opposite means low point. So f'x=(2/5)sin(x/5) with 0s at 0,pi,2pi,3pi,4pi.
Just before the first zero the slope is negative, so it is decreasing there, so 0 is a low point, and knowing sine pi will be a high point. So the concave down parts are 0,2pi,4pi...

it is increasing between 0 and pi, 2pi and 3pi, 4pi and 5pi...

Answer 2

derivative is sin(2x/5)/5 actually, but the method still applies