Find all relative extrema of the function. Use the second derlvative test where applicable. (If an answer does not exist, enter DNE.)

relative maximum
relative minimum
( )

Ask by Peters Murray. in the United States

Mar 22,2025

Question

Find all relative extrema of the function. Use the second derlvative test where applicable. (If an answer does not exist, enter DNE.)

relative maximum
relative minimum
( )

Ask by Peters Murray. in the United States

Mar 22,2025

UpStudy AI · Accepted Answer

1. **Compute the first derivative:**  
   $$
   f(x)=x^2+7x-2 \quad \Rightarrow \quad f'(x)=2x+7
   $$

2. **Solve for critical points:**  
   Set the derivative equal to zero:
   $$
   2x+7=0 \quad \Rightarrow \quad x=-\frac{7}{2}
   $$

3. **Compute the second derivative:**  
   $$
   f''(x)=2
   $$

4. **Apply the second derivative test:**  
   Since $$ f''(x)=2>0 $$ for all $$ x $$, the function is concave upward at the critical point $$ x=-\frac{7}{2} $$. Therefore, there is a relative minimum at this point.

5. **Find the $$ y $$-coordinate of the relative minimum:**  
   Substitute $$ x=-\frac{7}{2} $$ into $$ f(x) $$:
   $$
   f\left(-\frac{7}{2}\right)=\left(-\frac{7}{2}\right)^2+7\left(-\frac{7}{2}\right)-2=\frac{49}{4}-\frac{49}{2}-2
   $$
   Write all terms with a common denominator:
   $$
   f\left(-\frac{7}{2}\right)=\frac{49}{4}-\frac{98}{4}-\frac{8}{4}=\frac{49-98-8}{4}=\frac{-57}{4}
   $$
   
6. **Conclusion:**  
   - **Relative Maximum:** None (DNE)  
   - **Relative Minimum:** $$\left(-\frac{7}{2}, -\frac{57}{4}\right)$$
Relative maximum: DNE Relative minimum: $$\left(-\frac{7}{2}, -\frac{57}{4}\right)$$

Find all relative extrema of the function. Use the second derlvative test where applicable. (If an answer does not exist, enter DNE.)

relative maximum
relative minimum
( )

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Find all relative extrema of the function. Use the second derlvative test where applicable. (If an answer does not exist, enter DNE.) relative maximum relative minimum ( )

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Find all relative extrema of the function. Use the second derlvative test where applicable. (If an answer does not exist, enter DNE.)

relative maximum
relative minimum
( )