What are the local extrema, if any, of #f (x) =sqrt(4-x^2)#?
The extrema of f(x) is:
- Max of 2 at x = 0
- Min of 0 at x = 2, -2
To find the extrema of any function, you carry out the following:
- Differentiate the function
- Set the derivative equal to 0
- Solve for the unknown variable
- Substitute the solutions into f(x) (NOT the derivative)
- Differentiate the function:
By Chain Rule**:
Simplifying:
- Set the derivative equal to 0:
Now, since this is a product, you can set each part equal to 0 and solve:
- Solve for the unknown variable:
Now you can see that x = 0, and to solve the right side, raise both sides to the -2 to cancel out the exponent:
- Substitute the solutions into f(x):
I'm not going to write out the full solution for the substitution as it is straightforward, but I'll tell you:
Thus, you can see that there is an absolute maximum of 2 at x = 0, and an absolute minimum of 0 at x = -2, 2.
Hopefully everything was clear and concise! Hope I could help! :)
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To find the local extrema of ( f(x) = \sqrt{4 - x^2} ), we first need to find its critical points by taking the derivative and setting it equal to zero.
First, find the derivative of ( f(x) ): [ f'(x) = \frac{d}{dx}\sqrt{4 - x^2} = \frac{-x}{\sqrt{4 - x^2}} ]
Now, set the derivative equal to zero and solve for ( x ): [ \frac{-x}{\sqrt{4 - x^2}} = 0 ]
This equation is satisfied when ( x = 0 ).
Next, we need to check the endpoints of the domain of ( f(x) ), which is ([-2, 2]).
When ( x = -2 ), ( f(-2) = 0 ).
When ( x = 2 ), ( f(2) = 0 ).
So, the critical point is ( x = 0 ), and the function has local extrema at the endpoints ( x = -2 ) and ( x = 2 ).
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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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