Find and classify the absolute extrema of the function on the domain . If the function doesrrt have an absolute extremum, write Nione
for your answer.

Ask by Lawson Watkins. in the United States

Mar 27,2025

Question

Find and classify the absolute extrema of the function $$ K(z)=\sqrt{16-z^{2}} $$ on the domain $$ D=(-4,4) $$. If the function doesrrt have an absolute extremum, write Nione for your answer.

UpStudy AI · Accepted Answer

1. **Determine the domain and express the function:**  
   The function is  
   $$
   K(z)=\sqrt{16-z^2}
   $$  
   and the domain is  
   $$
   D=(-4,4).
   $$
   Note that the endpoints $$ z=-4 $$ and $$ z=4 $$ are not included in the domain.

2. **Find the derivative:**  
   Write the function as  
   $$
   K(z)=(16-z^2)^{1/2}.
   $$  
   Differentiate using the chain rule:  
   $$
   K'(z) = \frac{1}{2}(16-z^2)^{-1/2} \cdot (-2z) = \frac{-z}{\sqrt{16-z^2}}.
   $$

3. **Find the critical points:**  
   Set the derivative equal to zero:  
   $$
   \frac{-z}{\sqrt{16-z^2}} = 0.
   $$
   The fraction is zero when the numerator is zero (since $$\sqrt{16-z^2}$$ is positive for $$z$$ in $$(-4,4)$$). Thus,  
   $$
   -z = 0 \quad \Rightarrow \quad z = 0.
   $$

4. **Evaluate the function at the critical point:**  
   Compute $$ K(0) $$:  
   $$
   K(0)=\sqrt{16-0^2}=\sqrt{16}=4.
   $$

5. **Check the endpoints of the domain:**  
   The endpoints $$ z=-4 $$ and $$ z=4 $$ are not included in $$ D $$. Although  
   $$
   \lim_{z\to\pm4} K(z)=\sqrt{16-16}=0,
   $$  
   these values are not attained.

6. **Classification of extrema:**  
   - **Absolute Maximum:** The function attains the value $$4$$ at $$ z=0 $$, which is the absolute maximum on $$ D $$.  
   - **Absolute Minimum:** Since the infimum of the function on $$ D $$ is $$0$$ (approached as $$z$$ approaches $$-4$$ or $$4$$) but is never attained (the endpoints are not included), there is no absolute minimum.

**Final Answer:**

- Absolute maximum is $$4$$ at $$ z=0 $$.  
- There is no absolute minimum.
The function $$ K(z) = \sqrt{16 - z^2} $$ on the domain $$ D = (-4, 4) $$ has an absolute maximum of 4 at $$ z = 0 $$. There is no absolute minimum.

Find and classify the absolute extrema of the function on the domain . If the function doesrrt have an absolute extremum, write Nione
for your answer.

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Find and classify the absolute extrema of the function on the domain . If the function doesrrt have an absolute extremum, write Nione for your answer.