Problem 65 page 211
Given: We have a function f(x) = Ax2+Bx +C.
To prove: For any interval [a, b] the value of cgiven by the Mean Value
Theorem is the midpoint of the interval.
Demonstration: We need to show that it we choose c= (a+b)/2 then
f0(c) = f(b)−f(a)
b−a.(1)
Now f0(x) = 2Ax +Band so
f0(c) = f0µa+b
2¶
= 2Aµa+b
2¶+B
A(a+b) + B.
However,
f(b)−f(a)
b−a=Ab2+Bb +C−(Aa2+Ba +C)
b−a
=A(b2
−a2) + B(b−a)
b−a
A(b+a)(b−a) + B(b−a)
b−a
=A(a+b) + B,
proving 1.
Here is an example with f(x) = x2
−3x+ 2, a = 0, b = 4 and c= 2.
