Question

which interval has a positive average rate of change for $f(x)=-x^{2}+2x + 5$?
a. $1,2$
b. $0,3$
c. $2,3$
d. $0,1$

Explanation:

Step1: Recall the formula for average rate of change

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

Step2: Calculate the average rate of change for each interval

• For interval \([1,2]\):
• First, find \(f(1)\) and \(f(2)\).
• \(f(x)=-x^{2}+2x + 5\), so \(f(1)=-(1)^{2}+2\times(1)+5=-1 + 2+5 = 6\).
• \(f(2)=-(2)^{2}+2\times(2)+5=-4 + 4+5 = 5\).
• Then, the average rate of change is \(\frac{f(2)-f(1)}{2 - 1}=\frac{5 - 6}{1}=-1\).
• For interval \([0,3]\):
• Find \(f(0)\) and \(f(3)\).
• \(f(0)=-(0)^{2}+2\times(0)+5 = 5\).
• \(f(3)=-(3)^{2}+2\times(3)+5=-9+6 + 5 = 2\).
• The average rate of change is \(\frac{f(3)-f(0)}{3 - 0}=\frac{2 - 5}{3}=-1\).
• For interval \([2,3]\):
• \(f(2) = 5\) (calculated above), \(f(3)=2\).
• The average rate of change is \(\frac{f(3)-f(2)}{3 - 2}=\frac{2 - 5}{1}=-3\).
• For interval \([0,1]\):
• \(f(0) = 5\) (calculated above), \(f(1)=6\).
• The average rate of change is \(\frac{f(1)-f(0)}{1 - 0}=\frac{6 - 5}{1}=1\).

Answer:

d. \([0,1]\)