We know that the average rate of function f(x) from x=a to x=b i.e. in the interval [a,b] is calculated as follows:
[tex]Rate\ of\ change=\dfrac{f(b)-f(a)}{b-a}[/tex]
a)
[tex]f(x)=x^2+3x[/tex]
interval: [-2,3]
We have:
[tex]f(3)=3^2+3\times 3\\\\\\f(3)=9+9\\\\f(3)=18[/tex]
and
[tex]f(-2)=(-2)^2+3\times (-2)\\\\\\f(-2)=4-6\\\\\\f(-2)=-2[/tex]
The average rate of change is calculated as:
[tex]=\dfrac{f(3)-f(-2)}{3-(-2)}\\\\\\=\dfrac{18-(-2)}{5}\\\\\\=\dfrac{20}{5}\\\\\\=4[/tex]
Hence, Average rate of change is: 4
b)
[tex]f(x)=3x-8[/tex]
interval: [4,5]
[tex]f(5)=3\times 5-8\\\\\\f(5)=15-8\\\\\\f(5)=7[/tex]
and
[tex]f(4)=3\times 4-8\\\\\\f(4)=12-8\\\\\\f(4)=4[/tex]
The average rate of change is calculated as:
[tex]=\dfrac{f(5)-f(4)}{5-4}\\\\\\=\dfrac{7-4}{1}\\\\\\=3[/tex]
Hence, Average rate of change is: 3
c)
[tex]f(x)=x^2-2x[/tex]
interval : [-3,4]
[tex]f(4)=4^2-2\times 4\\\\\\f(4)=16-8\\\\\\f(4)=8[/tex]
[tex]f(-3)=(-3)^2-2\times (-3)\\\\\\f(-3)=9+6\\\\\\f(-3)=15[/tex]
The average rate of change is calculated as:
[tex]=\dfrac{f(4)-f(-3)}{4-(-3)}\\\\\\=\dfrac{8-15}{7}\\\\\\=\dfrac{-7}{7}=-1[/tex]
Hence, Average rate of change is: -1
d)
[tex]f(x)=x^2-5[/tex]
interval : [-1,1]
[tex]f(1)=1-5\\\\f(1)=-4[/tex]
and
[tex]f(-1)=(-1)^2-5\\\\\\f(-1)=1-5\\\\\\f(-1)=-4[/tex]
The average rate of change is calculated as:
[tex]=\dfrac{f(1)-f(-1)}{1-(-1)}\\\\\\=\dfrac{-4-(-4)}{2}\\\\\\=\dfrac{0}{2}=0[/tex]
Hence, Average rate of change is: 0
Hence, on arranging the functions from greatest to least value based on average rate of change in the specified interval is:
[tex]f(x)=x^2+3x<f(x)=3x-8<f(x)=x^2-5<f(x)=x^2-2x[/tex]
i.e. a < b < d < c
