`f(x) = 3 - |x - 3|, [0,6]` Determine whether Rolle’s Theorem can be applied to `f` on the closed interval `[a, b]`. If Rolle’s Theorem can be applied, find all values of `c` in the open...

`f(x) = 3 - |x - 3|, [0,6]` Determine whether Rolle’s Theorem can be applied to `f` on the closed interval `[a, b]`. If Rolle’s Theorem can be applied, find all values of `c` in the open interval

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Chapter 3, 3.2 - Problem 14 - Calculus of a Single Variable (10th Edition, Ron Larson).
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embizze | High School Teacher | (Level 1) Educator Emeritus

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Given f(x)=3-|x-3| on the interval [0,6]:

To apply Rolle's theorem the function must be continuous on the closed interval [a,b], differentiable on the open interval (a,b), and f(a)=f(b).

f is continuous everywhere, so it is continuous on the closed interval [0,6]. f(0)=f(6)=0.

However, the function is not differentiable at x=3. (The left and right hand derivatives disagree; there is a corner.)

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We cannot apply Rolle's theorem for this function on the given interval.

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The graph:

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