The graph of any quadratic function f(x) = ax2 + bx + c is a parabola. Prove tha

Question

The graph of any quadratic function f(x) = ax2 + bx + c is a parabola. Prove that the average 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. f(x) = ax2 + bx + c The slope of the tangent line at x = p is 2ap + b, the slope of the tangent line at x = q is 2aq + b, and the average of those slopes is (2aP + b) + (2aq + b)/2 The midpoint of the interval [p, q] is p + q/2 and the slope of the tangent line at the midpoint is 2a(P + q/2 ) + b = This is equal to ap + aq + b, as required.

Explanation / Answer

f(x)=ax^2 +bx+c

f'(x)=2ax +b <--------------------first blank

(2ap+b+2aq+b)/2 =ap+aq+b=a(p+q)+b<--------------------------second blank

slope of tangent at midpont =2a((p+q)/2) +b=a(p+q) +b --------------------->third blank

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