Answer
See the explanation below.
Work Step by Step
$F(x)$ is a polynomial, so it is continuous everywhere.
Observe the interval $[a,b]$.
Note that $\qquad f(a)=a,\qquad f(b)=b$.
$y_{0}=\displaystyle \frac{a+b}{2}$ is the arithmetic mean of $f(a)$ and $f(b)$, a number halfway between $f(a)$ and $f(b)$.
The Intermediate Value theorem guarantees that
there exists a c from the interval such that $f(c)=y_{0}=\displaystyle \frac{a+b}{2}.$
