Find x* on [0, 2] such that f(x*) equals the average value of f(x) = 5x + 9 over [0, 2].

Answer 1

Tha average value of f is the integral of the function (over that [0,2] interval) divided by the interval width, 2.

I assume you know how to integrate the function. The value of the integral is
5/2*4 + 18 = 28 . The average value is therefre 28/2 = 14

Now find what value of x that makes
5x + 9 = 14. It is 1

That should not be a surprise. For a linear function, the average is the value in the middle of the interval.