Question

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

Explanation:

Step1: Recall the formula for average rate of change

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

Step2: Calculate for option A \([-1,2]\)

First, find \(f(-1)\) and \(f(2)\).
\(f(-1)=(-1)^2 + 10=1 + 10 = 11\)
\(f(2)=2^2+10=4 + 10=14\)
Then, calculate the average rate of change: \(\frac{f(2)-f(-1)}{2-(-1)}=\frac{14 - 11}{3}=\frac{3}{3}=1>0\)

Step3: Calculate for option B \([-3,1]\)

Find \(f(-3)\) and \(f(1)\)
\(f(-3)=(-3)^2+10=9 + 10=19\)
\(f(1)=1^2+10=1+10 = 11\)
Calculate the average rate of change: \(\frac{f(1)-f(-3)}{1-(-3)}=\frac{11 - 19}{4}=\frac{-8}{4}=-2<0\)

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

Find \(f(-4)\) and \(f(-1)\)
\(f(-4)=(-4)^2+10=16 + 10=26\)
\(f(-1)=(-1)^2+10=1+10 = 11\)
Calculate the average rate of change: \(\frac{f(-1)-f(-4)}{-1-(-4)}=\frac{11 - 26}{3}=\frac{-15}{3}=-5<0\)

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

Find \(f(-3)\) and \(f(3)\)
\(f(-3)=(-3)^2+10=9 + 10=19\)
\(f(3)=3^2+10=9+10 = 19\)
Calculate the average rate of change: \(\frac{f(3)-f(-3)}{3-(-3)}=\frac{19 - 19}{6}=0\)

Answer:

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