Let f be continuous on [a,b] Show that there is a ξ ∈ [a,b]

Question

There exists a ξ such that $\textstyle \int$ $f(x)dx = f(ξ)(b − a)$.

Let $f (x) := 2x^2 − 5x − 6$ and [a, b] := [−1, 3] How do I determine ξ?

Answer 1

Hint: Write $\chi = \int_{-1}^3 f(x) dx $. Then $\chi = 4f(\xi) $ so $$ 2\xi^2-5\xi-6 = \frac{\chi}{4}. $$ So you'll be solving the quadratic $2\xi^2-5\xi-6 - \frac{\chi}{4}=0$ for $\xi$.