Question

on which interval does ( h(x) ) have an average rate of change of zero?
choose 1 answer!
(a) ( -5 leq x leq 0 )
(b) ( 5 leq x leq 7 )
(c) ( -9 leq x leq -5 )
(d) ( 0 leq x leq 5 )

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 \(\frac{h(b)-h(a)}{b - a}\). If the average rate of change is zero, then \(h(b)=h(a)\).

Step2: Analyze each interval

• For interval \(-5\leq x\leq0\):

Let \(a=-5\) and \(b = 0\). From the graph, \(h(-5)
eq h(0)\) ( \(h(-5)\) is approximately \(2\) and \(h(0)=7\)).

• For interval \(5\leq x\leq7\):

Let \(a = 5\) and \(b=7\). From the graph, \(h(5)
eq h(7)\) ( \(h(5)=-3\) and \(h(7)\) is between \(3\) and \(4\)).

• For interval \(-9\leq x\leq - 5\):

Let \(a=-9\) and \(b=-5\). From the graph, the function is constant over this interval. So, \(h(-9)=h(-5)\). Then \(\frac{h(-5)-h(-9)}{-5-(-9)}=\frac{h(-5)-h(-5)}{4}=0\).

• For interval \(0\leq x\leq5\):

Let \(a = 0\) and \(b = 5\). From the graph, \(h(0)=7\) and \(h(5)=-3\), so \(h(0)
eq h(5)\).

Answer:

C. \(-9\leq x\leq - 5\)