How do you verify that the hypotheses of rolles theorem are right for #f(x)= x sqrt(x+2)# over the interval [2,4]?
The hypotesis of the Rolles theorem are:
The derivative is:
So the theorem can't be applicated!
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To verify that the hypotheses of Rolle's Theorem are satisfied for ( f(x) = x \sqrt{x + 2} ) over the interval ([2, 4]), follow these steps:

Check Continuity: Verify that ( f(x) ) is continuous on the closed interval ([2, 4]).

Check Differentiability: Confirm that ( f(x) ) is differentiable on the open interval ((2, 4)).

Check Endpoints: Ensure that ( f(2) = f(4) ).

Calculate Derivative: Find the derivative of ( f(x) ), denoted as ( f'(x) ).

Find Critical Point: Determine if there exists a point ( c ) in the open interval ((2, 4)) such that ( f'(c) = 0 ).

Verify Hypotheses: Confirm that all conditions are met. If all conditions are satisfied, then Rolle's Theorem is applicable.

Conclusion: If all conditions are met, conclude that there exists at least one ( c ) in the open interval ((2, 4)) such that ( f'(c) = 0 ), implying that there exists a point ( c ) in the open interval ((2, 4)) such that ( f'(c) = 0 ).

Additional Note: Rolle's Theorem guarantees the existence of at least one point ( c ) in the open interval ((2, 4)) such that the derivative of ( f(x) ) is zero, given that the function meets the specified criteria.
If all steps are successfully completed, the hypotheses of Rolle's Theorem are verified for ( f(x) = x \sqrt{x + 2} ) over the interval ([2, 4]).
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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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