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Problem 80 Hard Difficulty

The graph of any quadratic function $ f(x) = ax^2 + bx + c $ is a parabola. Prove that the average of the slopes of the slopes of the tangent lines to the parabola at the endpoints of any interval $ [p, q] $ equals the slope of the tangent line at the midpoint of the interval.

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he is clear. So when you right here, so we have a for blacks is equal to x square plus b x plus c We're gonna differentiate it when we get to a X plus. Be Then we're gonna find this slope at the interval. PICO McCue the end points and we're going to average them together to a p plus D for Q well, to a Q plus B. We average them together when we get to a P plus B plus two es que plus B over too, which is, he looked to a P plus. Take you plus B, and we're going to find the slope at the midpoint of P comma Q to a P plus Q. Over too, must be the derivative of P plus Q. Over too is equal to a Q plus A people plus P. So here we have proved at the average of the slopes, the slopes of detention line to the proble, the end points of any interval. PICO McCue equals the slope of detention line at the midpoint of the interval