5-1 The Mean Value Theorem

 $f\left(x\right)=\left(x+3\right)\left(x-1\right)^2,\ \ \ \ \left[-3,1\right]$  

Find the value of c which is guaranteed by Rolle's Theorem.

The geometrical interpretation of Mean Value theorem for the curve  $y=f\left(x\right)$   defined in  $\left[a,b\right]$   says  $\in$   (there exists) ...

The function, F, above satisfies the conclusion of Rolle's Theorem in the interval [a,b] because:

I. F is continuous

II. F is differentiable on (a,b)

III. F(a) = F(b) = 0

The value of c guaranteed to exist by the MVT for  $f\left(x\right)=x^2$  on the interval [0,3] is:

 $f\left(x\right)=x^3+2x,\ \ \ \left[-1,1\right]$  


Find the value of c guaranteed by the Mean Value Theorem.

Does the mean value theorem apply to f on [0, 5] ?