# How do you determine whether Rolle’s Theorem can be applied to #f(x)=x(x-6)^2# on the interval [0,6]?

See the explanation section.

When we are asked whether some theorem "can be applied" to some situation, we are really being asked "Are the hypotheses of the theorem true for this situation?"

(The hypotheses are also called the antecedent, of 'the if parts'.)

So we need to determine whether the hypotheses ot Rolle's Theorem are true for the function

Rolle's Theorem has three hypotheses:

We say that we can apply Rolle's Theorem if all 3 hypotheses are true.

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To determine whether Rolle's Theorem can be applied to ( f(x) = x(x-6)^2 ) on the interval [0,6], you need to check the following conditions:

- ( f(x) ) is continuous on the closed interval [0,6].
- ( f(x) ) is differentiable on the open interval (0,6).
- ( f(0) = f(6) ).

If all three conditions are satisfied, then Rolle's Theorem can be applied to the function on the given interval.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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