# 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?

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 of 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.

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

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

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To determine if Rolle's Theorem can be applied to ( f(x) = x\sin(x) ) on the interval ([-4,4]), we need to check if the function meets the necessary conditions. Rolle's Theorem requires the function to be continuous on the closed interval ([a,b]), differentiable on the open interval ((a,b)), and have equal function values at the endpoints of the interval.

For ( f(x) = x\sin(x) ) on ([-4,4]), it is continuous and differentiable on ((-4,4)). Additionally, ( f(-4) = -4\sin(-4) \approx -2.74 ) and ( f(4) = 4\sin(4) \approx 2.74 ). Since ( f(-4) \neq f(4) ), Rolle's Theorem cannot be applied to this function on the interval ([-4,4]).

However, to find all values of ( c ) in the interval ([-4,4]) for which ( f'(c) = 0 ), we first find the derivative of ( f(x) = x\sin(x) ). The derivative is ( f'(x) = \sin(x) + x\cos(x) ).

Next, we set ( f'(x) ) equal to zero and solve for ( x ) to find critical points:

[ \sin(x) + x\cos(x) = 0 ]

This equation does not have simple closed-form solutions, so we typically use numerical methods or graphical methods to approximate the values of ( x ) where ( f'(x) = 0 ) within the interval ([-4,4]).

Using numerical methods or a graphing calculator, we can find the values of ( x ) (which correspond to ( c )) for which ( f'(c) = 0 ) within the interval ([-4,4]).

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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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- Given the function #f(x)=5sqrt(25-x^2)#, how do you determine whether f satisfies the hypotheses of the Mean Value Theorem on the interval [0,5] and find the c?
- Given the function #f(x) = 2x^2 − 3x + 1#, how do you determine whether f satisfies the hypotheses of the Mean Value Theorem on the interval [0,2] and find the c?
- Is #f(x)=-4x^3+4x^2+2x-1# increasing or decreasing at #x=2#?

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