x h(x) -5 6 -4 4 -3 2 -2 0 -1 -1 0 -2 1 -3 2 0 3 3 4 6 5 9 over which interval does h have a positive…

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function (y = h(x)) over the interval ([a,b]) is given by (\frac{h(b)-h(a)}{b - a}).

Step2: Calculate for option A ([-5,3])

Here (a=-5), (b = 3). (h(-5)=6), (h(3)=3). Then (\frac{h(3)-h(-5)}{3-(-5)}=\frac{3 - 6}{3 + 5}=\frac{-3}{8}<0).

Step3: Calculate for option B ([0,1])

Here (a = 0), (b=1). (h(0)=-2), (h(1)=-3). Then (\frac{h(1)-h(0)}{1-0}=\frac{-3-(-2)}{1}=\frac{-3 + 2}{1}=-1<0).

Step4: Calculate for option C ([-3,1])

Here (a=-3), (b = 1). (h(-3)=2), (h(1)=-3). Then (\frac{h(1)-h(-3)}{1-(-3)}=\frac{-3-2}{1 + 3}=\frac{-5}{4}<0).

Step5: Calculate for option D ([2,3])

Here (a = 2), (b=3). (h(2)=0), (h(3)=3). Then (\frac{h(3)-h(2)}{3-2}=\frac{3-0}{1}=3>0).

Answer:

D. ([2,3])