How do you determine whether Rolle’s Theorem can be applied to #f(x)=x(x-6)^2# on the interval [0,6]?
1 Answer
See the explanation section.
Explanation:
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:
H1 :
H2 :
H3 :
We say that we can apply Rolle's Theorem if all 3 hypotheses are true.
H1 : The function
H2 : The function
[Because the derivative,
H3 :
Therefore we can apply Rolle's Theorem to
Extra
Because the hypotheses are true, we know without further work, that the conclusion of Rolle's Theorem. That is, we know that there is a
Sometimes, as an equation solving exercise, students are asked to find the
In this question,