Consider the function f(x)=x^2-4x+5 on the interval [0,4].

Question

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

f(x) is _____ on [0,4];

f(x) is _____ on (0,4);

and f(0)=f(4)= ______

Then by Rolle's theorem, there exists a c such f'(c)=0

Find the value c.

c=_____

Answer 1

f(x) is a polynomial, so f is continuous and differentiable everywhere.

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

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

f(0)=f(4)=5

f'(x)=2x-4 which is zero when x=2

f'(2)=0, so c=2