6.5. Differentiable Functions

Examples 6.5.10(a):

Does Rolle's theorem apply to defined on (-3, 3) ? If so, find the number guarantied by the theorem to exist.
Your browser can not handle Java applets This function is continuous on the interval [-3, 3], and differentiable on (-3, 3). It is not differentiable at x = -3 and x = +3, but Rolle's theorem does not require the function to be differentiable at the endpoints. Also, f(3) = f(-3) = 0. Therefore, Rolle's theorem does apply.

It guaranties the existence of a number c between -3 and 3 such that f'(c) = 0. It does not specify where exactly this number is located. However, a quick calculation shows that the number c is in fact c = 0.

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