`f(x) = sec(x), [pi,2pi]` 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) = sec(x), [pi,2pi]` 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 22 - Calculus of a Single Variable (10th Edition, Ron Larson).
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sciencesolve | Teacher | (Level 3) Educator Emeritus

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The Rolle's theorem is applicable to the given function, only if the function is continuous and differentiable over the interval, and f(a) = f(b). Since all trigonometric functions are continuous and differentiable on R, hence, the given function is continuous and differentiable on interval. Now, you need to check if `f(pi) = f(2pi).`

`f(pi) =sec (pi)= 1/(cos pi) = 1/(-1) = -1`

`f(2pi) =sec 2pi = 1/(cos 2pi) = 1/1 = 1`

Since one of all the three conditions is not valid, `f(pi) != f(2pi)` , you cannot apply Rolle's theorem.

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