How many values of c satisfy the conclusion of the Mean Value Theorem for f(x) = x^3 + 1 on the interval [-1,1]?

1 Answer
Bill K.
Jun 17, 2015

There are two values of #c# that work.

Explanation:

The derivative of #f# is #f'(x)=3x^2#. The slope of the secant line is #(f(1)-f(-1))/(1-(-1))=(2-0)/2=1#. Setting #f'(x)=1# and solving for #x# results in #x=\pm 1/sqrt{3}#. These are the two "#c#" values, and they are in the interval #[-1,1]#.

Here's a picture for the given situation:

enter image source here

The red line is the secant line between #(-1,f(-1))=(-1,0)# and #(1,f(1))=(1,2)#. The green line is the tangent line to #f# at #(-1/sqrt(3),f(-1/sqrt(3)))\approx (-0.577,0.808)# and the gold line is the tangent line to #f# at #(1/sqrt(3),f(1/sqrt(3)))\approx (0.577,1.192)#. All these lines are parallel with a slope of 1.

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