Question

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 average rate of change? choose 1 answer: a -5,3 b 0,1 c -3,1 d 2,3

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. \([2,3]\)