Approximating Relative Minima or Maxima In 
Exercises 61 and 62, use a graphing utility to approximate 
(to two decimal places) any relative minima or maxima 
of the function.   61. f(x) = −x2 + 2x + 1

To find the relative minima or maxima of the function \( f(x) = -x^2 + 2x + 1 \), we can follow these steps:

### 1. **Find the Derivative:**
   The first step is to take the derivative of the function to find the critical points.
   \[
   f'(x) = \frac{d}{dx}(-x^2 + 2x + 1) = -2x + 2
   \]

### 2. **Set the Derivative to Zero:**
   To find the critical points, set the derivative equal to zero and solve for \( x \).
   \[
   -2x + 2 = 0
   \]
   \[
   2x = 2
   \]
   \[
   x = 1
   \]

### 3. **Determine if it’s a Minimum or Maximum:**
   To determine whether this critical point is a minimum or maximum, examine the second derivative:
   \[
   f''(x) = \frac{d}{dx}(-2x + 2) = -2
   \]
   Since the second derivative is negative (\( f''(x) < 0 \)), this means that the function has a **relative maximum** at \( x = 1 \).

### 4. **Calculate the Function Value:**
   Substitute \( x = 1 \) back into the original function to find the value of the relative maximum:
   \[
   f(1) = -(1)^2 + 2(1) + 1 = -1 + 2 + 1 = 2
   \]

### 5. **Conclusion:**
   The function \( f(x) = -x^2 + 2x + 1 \) has a relative maximum at \( x = 1 \) with a value of \( f(1) = 2 \).

If using a graphing utility, you would plot the function and identify the relative maximum point at approximately \( (1.00, 2.00) \).

Would you like to explore any further details or have any questions about this?

Here are some related questions:

1. How do you find the relative minima or maxima for a cubic function?
2. What does the second derivative tell us about the concavity of the function?
3. Can a function have more than one relative maximum or minimum?
4. How can you verify if a critical point is an inflection point?
5. What are the real-world applications of finding relative maxima and minima?

**Tip:** When using a graphing utility, zooming in near critical points can give you more accurate approximations for minima and maxima.

Learn how to approximate relative minima or maxima for the function f(x) = -x^2 + 2x + 1 using a graphing utility. This problem covers calculus concepts including derivatives, critical points, and the second derivative test. Suitable for Grades 11-12 students.

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