Question

1. which interval has a positive average rate of change for f(x) = -x² + 2x + 5?

a. 1,2
b. 2,3
c. 0,3
d. 0,1

Explanation:

Step1: Recall the average rate of change formula

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 for option a: \([1,2]\)

First, find \(f(1)\) and \(f(2)\).
For \(f(x)=-x^{2}+2x + 5\), \(f(1)=-(1)^{2}+2(1)+5=-1 + 2+5 = 6\), \(f(2)=-(2)^{2}+2(2)+5=-4 + 4+5 = 5\).
The average rate of change is \(\frac{f(2)-f(1)}{2 - 1}=\frac{5 - 6}{1}=-1\).

Step3: Calculate for option b: \([2,3]\)

Find \(f(2)\) and \(f(3)\). \(f(2) = 5\) (from above), \(f(3)=-(3)^{2}+2(3)+5=-9+6 + 5=2\).
The average rate of change is \(\frac{f(3)-f(2)}{3 - 2}=\frac{2 - 5}{1}=-3\).

Step4: Calculate for option c: \([0,3]\)

Find \(f(0)\) and \(f(3)\). \(f(0)=-(0)^{2}+2(0)+5 = 5\), \(f(3)=2\) (from above).
The average rate of change is \(\frac{f(3)-f(0)}{3 - 0}=\frac{2 - 5}{3}=\frac{-3}{3}=-1\).

Step5: Calculate for option d: \([0,1]\)

Find \(f(0)\) and \(f(1)\). \(f(0)=5\) (from above), \(f(1)=6\) (from above).
The average rate of change is \(\frac{f(1)-f(0)}{1 - 0}=\frac{6 - 5}{1}=1\).

Answer:

d. \([0,1]\)