How do you find the value of c that satisfy the equation #(f(b)-f(a))/(b-a)=f'(c)# in the conclusion of the mean value theorem for the function #f(x)=4x^2+4x-3# on the interval [-1,0]#?

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Answer 1 · 2016-12-26T23:10:35

Please see below.

For the problem under consideration, we have

#f(x) = 4x^2+4x-3#

#f'(x) = 8x+4#

#a = -1# and #b=0#.

So we need to solve the equation

#f'(x) = (f(0)-f(-1))/(0-(-1)) # on the interval #(-1,0)#

Solve

#8x+4 = (-3-(-3))/(0-(-1))#

#8x+4 = 0#

#x = -1/2# is in the interval #(-1,0)#.

So #c = -1/2#.