Question

f(x)=x^{2}+10
over which interval does f have a positive average rate of change?
choose 1 answer:
a -3,1
b -1,2
c -4,-1
d -3,3

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 for option A \([-3,1]\)

First, find \(f(-3)\) and \(f(1)\).
\(f(-3)=(-3)^{2}+10=9 + 10=19\), \(f(1)=1^{2}+10=1+10 = 11\).
The average rate of change is \(\frac{f(1)-f(-3)}{1-(-3)}=\frac{11 - 19}{1 + 3}=\frac{-8}{4}=-2\).

Step3: Calculate for option B \([-1,2]\)

Find \(f(-1)\) and \(f(2)\).
\(f(-1)=(-1)^{2}+10=1+10 = 11\), \(f(2)=2^{2}+10=4 + 10=14\).
The average rate of change is \(\frac{f(2)-f(-1)}{2-(-1)}=\frac{14 - 11}{2+1}=\frac{3}{3}=1\).

Step4: Calculate for option C \([-4,-1]\)

Find \(f(-4)\) and \(f(-1)\).
\(f(-4)=(-4)^{2}+10=16+10 = 26\), \(f(-1)=(-1)^{2}+10=1+10 = 11\).
The average rate of change is \(\frac{f(-1)-f(-4)}{-1-(-4)}=\frac{11 - 26}{-1 + 4}=\frac{-15}{3}=-5\).

Step5: Calculate for option D \([-3,3]\)

Find \(f(-3)\) and \(f(3)\).
\(f(-3)=(-3)^{2}+10=9+10 = 19\), \(f(3)=3^{2}+10=9 + 10=19\).
The average rate of change is \(\frac{f(3)-f(-3)}{3-(-3)}=\frac{19 - 19}{3+3}=0\).

Answer:

B. \([-1,2]\)