Find all relative extrema of the function. Use the second derivative test where applicable. (If an answer does not exist, enter DNE.)
f(x) = x2 + 7x − 5
relative maximum	(x, y)	=	

    

relative minimum	(x, y)	=	

    


Let's summarize the steps to find the relative extrema for \( f(x) = x^2 + 7x - 5 \).

### First Derivative
\[ f'(x) = 2x + 7 \]

### Critical Points
Set the first derivative equal to zero:
\[ 2x + 7 = 0 \]
\[ x = -\frac{7}{2} \]

### Second Derivative
\[ f''(x) = 2 \]

Since the second derivative is positive (\( f''(x) = 2 \) > 0), the function is concave up at \( x = -\frac{7}{2} \). Therefore, there is a relative minimum at this point.

### Value of the Function at the Critical Point
\[ f\left(-\frac{7}{2}\right) = \left(-\frac{7}{2}\right)^2 + 7\left(-\frac{7}{2}\right) - 5 \]
\[ = \frac{49}{4} - \frac{49}{2} - 5 \]
\[ = \frac{49}{4} - \frac{98}{4} - \frac{20}{4} \]
\[ = \frac{49 - 98 - 20}{4} \]
\[ = \frac{-69}{4} \]

### Relative Maximum
Since the second derivative test shows the function is concave up at the critical point, there is no relative maximum. Therefore, the answer for the relative maximum is DNE (Does Not Exist).

### Relative Minimum
The relative minimum point is:
\[ \left( -\frac{7}{2}, -\frac{69}{4} \right) \]

### Conclusion
- Relative Maximum: \((x, y) = \text{DNE}\)
- Relative Minimum: \((x, y) = \left( -\frac{7}{2}, -\frac{69}{4} \right)\)

If you have any questions or need more details, feel free to ask!

Here are some additional questions you might consider:
1. How do you find the critical points of a function?
2. What is the second derivative test?
3. How do you interpret the concavity of a function?
4. Can you explain the significance of relative extrema in a real-world context?
5. What other methods can be used to find relative extrema besides the second derivative test?

**Tip:** When verifying the nature of critical points, always check the sign of the second derivative to determine if the point is a relative minimum or maximum.

Discover how to find all relative extrema of the function f(x) = x^2 + 7x - 5 using the second derivative test where applicable. Learn about critical points, concavity, and the significance of relative extrema in calculus. Suitable for advanced high school students.

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