Engineering Board Exam Reviewer · Rolle's Theorem and the Mean Value Theorem · hard

Which function does NOT satisfy the hypotheses of Rolle's Theorem on [-1,1]?

1. f(x)=x2-1
2. f(x)= π x/2
3. f(x)=|x|
4. f(x)=1-x4

Answer

C. f(x)=|x|

Solution

Rolle's Theorem needs continuity on [-1,1], differentiability on (-1,1), and f(-1)=f(1). f(x)=|x| has f(-1)=f(1)=1 but is not differentiable at x=0. The other three are smooth and equal 0 at both endpoints.

More Rolle's Theorem and the Mean Value Theorem questions

1. Find the value of c guaranteed by the Mean Value Theorem for f(x)= x on [1,9].
2. Find the value of c guaranteed by the Mean Value Theorem for f(x)=x3 on [0,2].
3. Find the value of c guaranteed by Rolle's Theorem for f(x)=x2-4x+3 on [1,3].

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