Given the function #f(x) = x² - 3x + 1#, how do you determine whether f satisfies the hypotheses of the Mean Value Theorem on the interval [-1,1] and find the c?

1 Answer
Dec 24, 2016

See below.

Explanation:

You determine whether it satisfies the hypotheses by determining whether #f(x) =x^2-3x+1# is continuous on the interval #[-1,1]# and differentiable on the interval #(-1,1)#. (Those are the hypotheses of the Mean Value Theorem)

You find the #c# mentioned in the conclusion of the theorem by solving #f'(x) = (f(1)-f(-1))/(1-(-1))# on the interval #(-1,1)#.

#f# is a polynomial, so it is continuous on its domain, which includes #[-1,1]#

#f'(x) = 2x-3# which exists for all #x# so it exists for all #x# in #(-1,1)#

Therefore this function satisfies the hypotheses of the Mean Value Theorem on this interval.

To find #c# solve the equation #f'(x) = (f(1)-f(-1))/(1-(-1))#. Discard any solutions outside #(-1,1)#.

You should get #c = 0#.