Question

Calculus help! Using the mean value theorem!

Answer 1

`f` is differentiable across its domain, so the MVT says there is some value of
`c` in the open interval
`(a,b)` such that


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

You have


`f(x)=Ax^2+Bx+C\implies f'(x)=2Ax+B`

so the equation above becomes


`2Ac+B=((Ab^2+Bb+C)-(Aa^2+Ba+C))/(b-a)`

Solve for
`c`.


`2Ac+B=(A(b^2-a^2)+B(b-a))/(b-a)`

`2Ac+B=A(b+a)+B`

`2Ac=A(b+a)`

`c=\frac{b+a}2`

so
`c` is indeed the average of the endpoints, i.e. the midpoint.