Answer
$$f'\bigg(\frac{p+q}{2}\bigg) = \frac{f'(p)+f'(q)}{2}$$
Work Step by Step
$f(x) = ax^{2} + bx + x$
Use power rule to determine the derivative of the parabola equation.
$f'(x) = 2ax +b$
Find the slopes at the endpoints of the inteval [p,q] and average them.
$f'(p) = 2ap+b$
$f'(q) = 2aq+b$
$m_{a\nu e} = \frac{2ap+b+2aq+b}{2} = ap+aq+b$
Find the slope at the midpoint of the interval [p,q]
$f'\bigg(\frac{p+q}{2}\bigg) = 2a\bigg(\frac{f'(p)+f'(q)}{2}\bigg)+b$
$f'\bigg(\frac{p+q}{2}\bigg) = aq+ap+b$
$f'\bigg(\frac{p+q}{2}\bigg) = \frac{f'(p)+f'(q)}{2}$
