To find the maximum, minimum, and points of inflection of the function f(x) = -x³ + 3x² + 9x + 1, you need to follow these steps:
1. **Find the Critical Points:**
To find the critical points, you need to take the derivative of the function and set it equal to 0 to find where the slope is 0.
f'(x) = -3x² + 6x + 9
Set f'(x) = 0 and solve for x:
-3x² + 6x + 9 = 0
You'll get the critical points by solving this quadratic equation.
2. **Determine the Nature of Critical Points:**
Use the second derivative test to determine whether each critical point is a maximum, minimum, or a point of inflection.
Find f''(x) by taking the derivative of f'(x).
If f''(x) > 0, it's a minimum.
If f''(x) < 0, it's a maximum.
If f''(x) = 0, it's a point of inflection.
3. **Calculate Maximum and Minimum:**
Plug the critical points into the original function to find the corresponding y-values. The highest y-value is the maximum, and the lowest is the minimum.
4. **Identify Points of Inflection:**
To find points of inflection, set the second derivative f''(x) = 0 and solve for x. Plug these x-values into the original function to get the corresponding y-values.
By following these steps, you'll be able to determine the maximum, minimum, and points of inflection of the function f(x) = -x³ + 3x² + 9x + 1 accurately.
