Let g and h be any two twice-differentiable functions that are defined for all real numbers and that satisfy the following properties for all x:

(I) $\displaystyle (g(x))^2$+$\displaystyle (h(x))^2$=1

(II)$\displaystyle g'(x)=(h(x))^2$

(III)$\displaystyle h(x)>0$

(IV)$\displaystyle g(0)=0$

(a)Prove that $\displaystyle h'(x)=-g(x)h(x) $ for all x

(b)Prove that h has a relative maximum at x=o

(c)Prove that the graph of g has a point of inflection at x=o