`f(x) = sin(3x), [0,(pi/3)]` 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) = sin(3x), [0,(pi/3)]` 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 19 - Calculus of a Single Variable (10th Edition, Ron Larson).
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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 polynomial functions are continuous and differentiable on R, hence, the given function is continuous and differentiable on interval. Now, you need to check if `f(0) = f(pi/3).`

`f(0) = sin(3*0) = 0`

`f(3) = sin(3*(pi)/3) = sin pi = 0`

Since all the three conditions are valid, you may apply Rolle's theorem:

` f'(c)(b-a) = 0`

Replacing `pi/3 ` for b and 0 for a, yields:

`f'(c)(pi/3 - 1) = 0`

You need to evaluate f'(c), using chain rule:

`f'(c) = (sin(3c))' = (cos 3c)*(3c)' = 3cos 3c`

 `f'(c) = 3cos 3c`

Replacing the found values in equation `f'(c)(pi/3 - 1) = 0.`

`(3cos 3c)(pi/3 - 1) = 0 => cos 3c = 0 => 3c = pi/2 or 3c = 3pi/2`

`c = pi/6 or c = pi/2`

Since `c = pi/2 ` does not belong to `(0,pi/3), ` only` c = pi/6`  is a valid value.

Hence, in this case, the Rolle's theorem may be applied for `c = pi/6.`

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