SOLUTION: Can we apply role's theorem in the interval [-2, 2] for following functions: f(x) = 4.x^4 and g(x) = πx?

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Question 1184172: Can we apply role's theorem in the interval [-2, 2] for following functions:
f(x) = 4.x^4 and g(x) = πx?
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

Rolle’s Theorem - Let f%28x%29 be continuous on the closed interval [a, b] and differentiable on the interval [a, b].
If f%28+a%29+=f%28b%29 then there is at least one number c in [a, b] such that f' %28c%29=0.

f%28x%29+=+4x%5E4+ in the interval [-2,+2]

check:
if continuous on [-2,+2]
if differentiable on [-2,+2]
if f%28-2%29+=f%282%29

The function f%28x%29+=4x%5E4 is continuous and differentiable on [-2,+2].
f%28-2%29+=+4%28-2%29%5E4=64
f%282%29+=+4%282%29%5E4=64 =>f%28-2%29+=f%282%29
=> role's theorem can be applied
f'%28x%29+=+16x%5E3
f'+%28c%29=0 gives 16x%5E3=0 => c=0

and
g%28x%29+=+pi%2Ax
is continuous and differentiable on [-2, 2]
g%28-2%29+=+pi%2A%28-2%29=-2pi
g%282%29+=+pi%2A%282%29=2pi
g'%28x%29+=pi
=> since g%28-2%29+%3C%3Eg%282%29 , role's theorem cannot be applied