# Math Help - Help with the mean value theorem

1. ## Help with the mean value theorem

I went in to my professor for "extra help" on this problem and left more confused than before I went in.

Can someone on here try to explain this to me!

Question:

For what values of a, m, and b does the function satisfy the hypotheses of the Mean Value Theorem on the inverval [0, 2]

Piecewise function:
f(x)= 3, x=0
f(x)=-x^2 + 3x + a, 0<x<1
f(x)=mx + b 1</= x </= 2

Thank you so much for any help!

2. Originally Posted by KarlosK
I went in to my professor for "extra help" on this problem and left more confused than before I went in.

Can someone on here try to explain this to me!

Question:

For what values of a, m, and b does the function satisfy the hypotheses of the Mean Value Theorem on the inverval [0, 2]

Piecewise function:
f(x)= 3, x=0
f(x)=-x^2 + 3x + a, 0<x<1
f(x)=mx + b 1</= x </= 2

Thank you so much for any help!
In order for the mean value theorem to be applicable, f(x) needs to be continuous on the entire closed interval [0;2], and differentiable in its interior ]0;2[.
Continuity at x=0 requires a=3, and you can determine the requisite values of b and m from the tangent to the graph of $y=-x^2+3x+3$ at $x_0=1$. Because, for f(x) to be differentiable at $x_0=1$ the graph $y=mx+b$ must be that tangent itself...