Given the function #f(x)=x^3-9x#, how do you determine whether f satisfies the hypotheses of the Mean Value Theorem on the interval [-1,1] and find the c?

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Answer 1 · 2016-11-19T14:53:57

The values of c, are #=1# and #=-1#

#f(x)# is a polynomialis which is continuous on the interval #[-1,1]# and differentiable on # ] -1,1 [#

#f(-1)=(-1)^3-9(-1)=+6#

#f(1)=1^3-9=-6#

Therefore, #EE# c#in ] -1,1 [#

such that #f'(c)=(f(b)-f(a))/(b-a)#

#f'(c)=(-6-6)/2=-6#

We must calculate,

#f'(x)=3x^2-9#

#3x^2-9=-6#

3x^2=9-6=3#

#x=+-1#

Therefore #c=1# or #c=-1#

#c in [-1,1]#

So, the mean value theorem is verified