Verify that this function satisfies the three hypotheses of Rolle's Theorem on the inverval.

Question

Consider the function f(x)=x²-4x+2 on the interval [0,4],

f(x) is ____ on [0,4],

f(x) is ____ on (0,4)

and f(0)=f(4)=2

Answer 1

Based on Rolle's Theorem,

f(x) is continuous on [0,4].

f(x) is differentiable on (0,4).

and f(0) = f(4) = 2.

There exists some value c within the interval that the following is true:

f'(c) = 0.

f'(x) = 2x - 4

f'(c) = 2c - 4 = 0

∴ c = 2

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