How do you find local maximum value of f using the first and second derivative tests: #1/3(y-2)=sin1/2(x-90*) #?
See the answer below:
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To find the local maximum value of ( f ) using the first and second derivative tests for the function ( \frac{1}{3}(y-2) = \sin\left(\frac{1}{2}(x-90^\circ)\right) ), follow these steps:
- Find the first derivative ( f'(x) ) by implicitly differentiating the given function with respect to ( x ).
- Set ( f'(x) ) equal to zero and solve for ( x ) to find critical points.
- Find the second derivative ( f''(x) ).
- Evaluate ( f''(x) ) at each critical point found in step 2.
- Use the first and second derivative tests to determine the nature of each critical point and identify local maximum values.
Let's go through the steps:
-
Implicitly differentiate ( \frac{1}{3}(y-2) = \sin\left(\frac{1}{2}(x-90^\circ)\right) ) with respect to ( x ) to find ( f'(x) ). [ f'(x) = \frac{d}{dx}\left(\sin\left(\frac{1}{2}(x-90^\circ)\right)\right) ]
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Set ( f'(x) = 0 ) and solve for ( x ) to find critical points.
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Find the second derivative ( f''(x) ). [ f''(x) = \frac{d^2}{dx^2}\left(\sin\left(\frac{1}{2}(x-90^\circ)\right)\right) ]
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Evaluate ( f''(x) ) at each critical point found in step 2.
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Use the first and second derivative tests to determine the nature of each critical point and identify local maximum values.
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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.
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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