How do we use Rolle's Theorem to find whether function #f(x)=4x^2-8x+7# has a point where #f'(x)=0# in the interval #[-1,3]#?

1 Answer
Shwetank Mauria · Ratnaker Mehta
Mar 31, 2017

#x=1#, for details please see below.

Explanation:

Rolle's Theorem when applied the function #f(x)# must be continuous for #x# in the given range, here #[-1,3]# and #f(x)# must be differentiable for #x# in #(-1,3)#.

Here we have #a=-1# and #b=3# and as #f(x)=4x^2-8x+7#, we have #f(-1)=19# and #f(3)=19# and hence #f(a)=f(b)#.

Now according to Rolle's Theorem, if in such a function #f(a)=f(b)#, there is one #c#, where #f'(c)=0#

As #f'(x)=8x-8# and as #8x-8=0=>x=1#, we have at #x=1#, #f'(1)=0#

and our #c# is #1#.

graph{(y-4x^2+8x-7)((x+1)^2+(y-19)^2-0.01)((x-3)^2+(y-19)^2-0.01)((x-1)^2+(y-3)^2-0.015)=0 [-4, 4, -5, 23]}

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