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Whitepunk [10]
3 years ago
5

Calculus help! Using the mean value theorem!

Mathematics
1 answer:
marta [7]3 years ago
8 0
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)=\dfrac{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=\dfrac{(Ab^2+Bb+C)-(Aa^2+Ba+C)}{b-a}

Solve for c.

2Ac+B=\dfrac{A(b^2-a^2)+B(b-a)}{b-a}
2Ac+B=A(b+a)+B
2Ac=A(b+a)
c=\dfrac{b+a}2

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