Here we need to find the average rate of change of [tex]f(x)[/tex].
Now,
[tex]f(x)=-2x^2+4x+2[/tex]
Let us find out the value of [tex]f(x)[/tex] at the point [tex]x=1.2[/tex].
Plugging [tex]x=1.2[/tex] in the equation we get:
[tex]f(1.2)=-2(1.2)^2+4(1.2)+2=-2\times 1.44+4.8+2=-2.88+4.8+2=3.92[/tex]
So,
[tex]f(1.2)=3.92[/tex]
Now calculating the value of [tex]f(x)[/tex] at [tex]x=3.8[/tex].
[tex]f(3.8)=-2(3.8)^2+4(3.8)+2=-2(14.44)+15.2+2=-28.88+15.2+2=-11.68[/tex]
So,
[tex]f(3.8)=-11.68[/tex]
Now that we have the values of the function at two distinct points, we can find the average by using the formula given below:
[tex]Avg= \frac{sum}{n}[/tex], where 'n' represents the number of values and that is two in our case and sum represents the sum of the values of the function.
Therefore,
[tex]Avg= \frac{3.92+(-11.68)}{2} =\frac{-7.76}{2} =-3.88[/tex]
So, the average rate of change of the function [tex]f(x)[/tex] from 1.2 to 3.8 is [tex]-3.88[/tex].
