Question

Decide whether Rolle's Theorem can be applied to f(x) = x^2 − 3 on the interval [–2, 2]. If Rolle's Theorem can be applied, find all value(s) of c in the interval such that f '(c) = 0?

Answer 1

`"Rolle's theorem can be applied"`
`"c = 0 is the only value such that f'(c) = 0"`

Explanation:

`"To apply Rolle in an interval [a, b], f(a) must be equal to f(b)."`
`"Here we have f(-2) = f(2) = 1, because (-2)²-3 = (2)² -3 = 1."`
`"So f(a)=f(b), so we can apply Rolle"`
`"Rolle says that there exists a value c in that interval so that"`
`"f'(c) = 0."`
`"So here f '(x) = 2 x."`
`"So 2 x = 0 , if and only if x = 0."`
`"So c = 0 is the value we search for as it lays in [-2 , 2]."`