Calculus 1- Mean Value theorem Question

Question

show that the point c between a and b guaranteed by the mean value theorem is always a+b/2 for any a, b, and any quadratric function.

Answer 1

A generic quadratic function can be defined with the constants C1, C2, and C3:

f(x) = C1 x² + C2 x + C3

Let's first evaluate the derivative of the function:

f'(x) = 2*C1 x + C2

Now let's find the secant slope guaranteed by the Mean Value Theorem, which is defined by the total vertical change divided by the total horizontal change:

f'(c) = (f(b) - f(a)) / (b - a)

f'(c) = ((C1*b² + C2*b + C3) - (C1*a² + C2*a + C3)) / (b - a)

= (C1*(b²-a²) + C2*(b-a)) / (b-a)

= (C1*(b+a)(b-a) + C2*(b-a)) / (b-a) --> cancel (b-a) on the top and bottom:

= C1*(a + b) + C2

This problem suggests that for any quadratic function, the point c is always guaranteed to be (a + b)/2. Therefore, let's evaluate f'(c) where c = (a + b)/2.

f'(c) = 2*C1*(a + b)/2) + C2

= C1*(a + b) + C2