Find the average value of the function

Question

f(x)=8x^2-5x+6 , on the interval [3,5]. Find the value of x-coordinate at which the function assumes it's average value.

what is the average value = to ?
what is the x coordinate = to ?

Thanks

Answer 1

Let f(x)=8x^2-5x+6
The average value is the definite integral divided by the interval.
I=∫f(x)dx
=∫(8x^2-5x+6)dx
=[(8/3)x^3-(5/2)x^2+6x] from 3 to 5
=700/3

Average value, a
= I/(5-3)
= 350/3

To find x where f(x)=a, solve for x in
f(x) = 8x^2-5x+6 = 350/3
However, out of the two solutions, x=-3.42 and x=4.045, you will report the solution which is within the interval [3,5] and reject all solutions outside of the interval.