How do you find all numbers c in the interval show that the function #f(x)=x^3+x+1# satisfies the mean value theorem on the interval [-1,2]?

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Answer 1 · 2015-04-03T19:07:18

The secant of #f(x)=x^3+x+1# over the interval #[-1,2]#
is
#(f(2) - f(-1))/(2- (-1))#
#= (11- (-1))/(2+1) = 4#

According to the Mean Value Theorem there should exist at least one value #c# such that
#f'(c) = 4# for this interval.

Since #f'(x) = 3x^2+1#
we are looking for one (or more values) that satisfy
#3c^2+1=4#

#rarr c = +-1#

Within the interval #[-1,2]# the Mean Value Theorem is satisfied by
#c=-1# and #c=1#