Using the principle of the mean-value theorem on the indicated interval, how do you find all numbers c that satisfy the conclusion of the theorem #f(x) = x³ + 5x² - 2x - 5#; [-1, 2]?

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Answer 1 · 2016-12-18T07:42:35

The answer is # c=2/3#

The function #f(x)# is a polynomial continuous, defifined and differentiable over #RR#. So we can apply the mean value theorem.

#f(x)=x^3+5x^2-2x-5#

#f(-1)=-1+5+2-5=1#

#f(2)=8+20-4-5=19#

By the mean value theorem,

#f'(c)=(f(2)-f(-1))/(2--1)=(19-1)/(3)=6#

Also,

#f'(x)=3x^2+10x-2#

#:.f(c)=3c^2+10c-2#

But, #f'(c)=6#

Therefore,

#3c^2+10c-2=6#

#3c^2+10c-8=0#

#(3c-2)(c+4)=0#

So,

#c=2/3# or #c=-4#

#c=-4 !in [-1,2] #

But only #c=2/3 in [-1,2] #