Question

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$g(x)=-\frac{x^{2}}{4}+7$
over which interval does $g$ have a negative average rate of change?
choose 1 answer:
a $-2,0$
b $-4,-2$
c $0,4$
d $-8,-4$

Explanation:

Step1: Recall the formula for average rate of change

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

Step2: Calculate the average rate of change for option A

For the interval \([-2,0]\):
\(g(-2)=-\frac{(-2)^{2}}{4}+7=-\frac{4}{4}+7=- 1 + 7=6\)
\(g(0)=-\frac{0^{2}}{4}+7 = 7\)
The average rate of change is \(\frac{g(0)-g(-2)}{0-(-2)}=\frac{7 - 6}{2}=\frac{1}{2}>0\)

Step3: Calculate the average rate of change for option B

For the interval \([-4,-2]\):
\(g(-4)=-\frac{(-4)^{2}}{4}+7=-\frac{16}{4}+7=-4 + 7 = 3\)
\(g(-2)=-\frac{(-2)^{2}}{4}+7=-1 + 7=6\)
The average rate of change is \(\frac{g(-2)-g(-4)}{-2-(-4)}=\frac{6 - 3}{2}=\frac{3}{2}>0\)

Step4: Calculate the average rate of change for option C

For the interval \([0,4]\):
\(g(0)=-\frac{0^{2}}{4}+7=7\)
\(g(4)=-\frac{4^{2}}{4}+7=-4 + 7=3\)
The average rate of change is \(\frac{g(4)-g(0)}{4 - 0}=\frac{3-7}{4}=\frac{-4}{4}=-1<0\)

Step5: Calculate the average rate of change for option D

For the interval \([-8,-4]\):
\(g(-8)=-\frac{(-8)^{2}}{4}+7=-\frac{64}{4}+7=-16 + 7=-9\)
\(g(-4)=-\frac{(-4)^{2}}{4}+7=-4 + 7=3\)
The average rate of change is \(\frac{g(-4)-g(-8)}{-4-(-8)}=\frac{3-(-9)}{4}=\frac{12}{4}=3>0\)

Answer:

C. \([0,4]\)