Question

How do you determine if rolles theorem can be applied to `f(x)=xsinx` on the interval [-4,4] and if so how do you find all the values of c in the interval for which f'(c)=0?

Answer 1

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 of Rolle's Theorem are true for the function

`f(x) = xsinx` on the interval `[-4,4]`

Rolle's Theorem has three hypotheses:

H1 : `f` is continuous on the closed interval `[a,b]`
H2 : `f` is differentiable on the open interval `(a,b)`.
H3 : `f(a)=f(b)`

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

H1 : The function `f` in this problem is continuous on `[-4,4]` [Because, this function is a product of the identity function with the sine function. Bothe of those are continuous at every real number, and the product of continuous function is continuous. So this `f` is continuous at every real number. Therefore `f` if continuous on `[-4,4]`.]

H2 : The function `f` in this problem is differentiable on `(-4,4)`
[Because the derivative, `f'(x) = sinx+xcosx` exists for all real `x`. In particular, it exists for all `x` in `(-4,4)`.)

H3 : `f(-4) = (-4)sin(-4) = (-4)(-sin4) = 4sin4 = f(4)`

Therefore we can apply Rolle's Theorem to `f(x) = xsinx` on the interval `[-4,4]`. (Meaning "the hypotheses are true.)

To find the values of `c`, we need to solve `f'(x) = 0` and select the solutions in `(-4,4)`.

So we need to solve `sinx+xcosx = 0`

"By inspection `0` is a solution in the interval `(-4,4)`.

The graphing utility shows that there are 2 more solutions in the interval.

graph{sinx+xcosx [-4.384, 4.385, -2.19, 2.194]}