`f(x) = -x^2 + 3x, [0,3]` 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) = -x^2 + 3x, [0,3]` 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 9 - 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(3).

`f(0) = -0^2 + 3*0= 0`

`f(3) =-3^2 + 3*3 = 0`

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

`f'(c)(b-a) = 0`

Replacing 3 for b and 0 for a, yields:

`f'(c)(3-0) = 0`

You need to evaluate f'(c):

`f'(c) = (-c^2 + 3c)' => f'(c) = - 2c + 3`

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

`3(-2c + 3) = 0 => -6c + 9 = 0 => -6c = -9 => c = 3/2 in (0,3)`

Hence, in this case, the Rolle's theorem may be applied for `c = 3/2.`

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