The naswers are what are marked with a green check mark or s

The naswers are what are marked with a green check mark or star. However, i don\'t understand how to get these answers. Please write out fully and explain your answer for all stars! THank you!!!

Solution

y\'= 4/9 x^(-5/9)
y\'\'= -20/81 x^(-14/9)

For 1) set the first derivative =0
4/(9x^5/9)= 0
No zeros but extrema can still be located where first derivative does not exist, at x=0
When x= -1 f(x) <0 and when x=1 f(x)>0 so f(x) must be a minimum at x=0.

Second derivative does not exist at x=0 so it is a possible inflection point.
However, f\'\'(-1)<0 and f\'\'(1) <0 so the function does not change concavity at x=0 so x=0 is not an inflection point.