How do you use Rolle's Theorem on a given function #f(x)#, assuming that #f(x)# is not a polynomial?
I know how to apply the theorem for polynomials, but I am unsure on how to apply it for other types of function.
I know how to apply the theorem for polynomials, but I am unsure on how to apply it for other types of function.
See below.
To use Rolle's Theorem on any kind of function we need
The only real differences between applying Rolle's to polynomial and nonpolynomial functions are
it will take more to explain why we think the function does or does not satisfy the conditions for conituity and differentiabiity.
I'm not sure that this will fully answer your question, but I con't explain more without some more questions from you.
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Here are some examples.
For example:
Second example (same function)
That is enough to tell us that we cannot use Rolle's Theorem for this function on this interval.
Third example (different function)
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To use Rolle's Theorem on a given function (f(x)), which is not necessarily a polynomial, follow these steps:

Verify the conditions of Rolle's Theorem:
 The function (f(x)) must be continuous on the closed interval ([a, b]).
 The function (f(x)) must be differentiable on the open interval ((a, b)).
 The function must satisfy (f(a) = f(b)).

Identify the interval ([a, b]) where you want to apply Rolle's Theorem. Ensure that (f(x)) satisfies the conditions above for this specific interval.

Apply Rolle's Theorem: If (f(x)) meets the conditions, then there exists at least one (c) in the open interval ((a, b)) such that the derivative (f'(c) = 0).

Find (f'(x)): Differentiate (f(x)) to find (f'(x)). This step does not change whether (f(x)) is a polynomial or not; apply the differentiation rules as per the function's form.

Solve (f'(c) = 0): Solve the equation (f'(c) = 0) to find the value(s) of (c) in the interval ((a, b)). This will give you the point(s) at which the function's tangent is horizontal, as guaranteed by Rolle's Theorem.

Conclusion: The value(s) of (c) found in step 5 are the points that satisfy the conclusion of Rolle's Theorem for the given function (f(x)) on the interval ([a, b]).
Note: The process is the same for polynomial and nonpolynomial functions, with the key step being to ensure the function meets the theorem's conditions and then solving for when its derivative equals zero within the given interval.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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