So, you calculate the derivative of f, calculate the slope of the secant line between (a, f(a)) and (b, f(b)), set them equal to each other and solve for x. This process can be easily automated on the TI-89 using the function:
Define mvt(f, x, a, b) = solve(d(f,x)=((f|x=b)-(f|x=a))/(b-a),x)|(x>a and x <b)
This can be a very handy way to check your work, but like just about all calculator functions, blindly copying the results displayed on the calculator screen onto your paper can be hazardous to your grade - be sure that you read and understand the limitations below.
A solution for problem #28 on p. 168 of the Larsen text is shown at right (Find all values of c in the interval [-1, 1] that satisfy the Mean Value Theorem for the function f(x) = x(x2 - x - 2)). Note that the function f has 2 zeros, but only x = -1/3 is in the required interval.
At right, a graph of the function is shown with a tangent line at x = -1/3. The graph window is -1 < x < 1, and -2.5 < y < 1.