To find the relative and absolute extrema of the function f(x) = x³ – 6x² + 9x + 2, we will follow a systematic approach involving first and second derivatives.
Step 1: Find the first derivative
The first step is to find the first derivative of the function, which will help us to determine the critical points. The first derivative, denoted as f'(x), is calculated as follows:
f'(x) = 3x² - 12x + 9Step 2: Set the first derivative to zero
Next, we set the first derivative equal to zero to find the critical points:
3x² - 12x + 9 = 0Dividing by 3 gives:
x² - 4x + 3 = 0This can be factored into:
(x - 3)(x - 1) = 0Thus, the critical points are:
- x = 1
- x = 3
Step 3: Find the second derivative
To determine whether these critical points are relative maxima, minima, or neither, we will compute the second derivative:
f''(x) = 6x - 12Step 4: Evaluate the second derivative at critical points
We evaluate the second derivative at the critical points found in Step 2:
- For x = 1:
f''(1) = 6(1) - 12 = -6This indicates a relative maximum.
- For x = 3:
f''(3) = 6(3) - 12 = 6This indicates a relative minimum.
Step 5: Calculate function values at critical points
Now we calculate the function values at the critical points to find relative extrema:
- f(1) = (1)³ – 6(1)² + 9(1) + 2 = 1 – 6 + 9 + 2 = 6
- f(3) = (3)³ – 6(3)² + 9(3) + 2 = 27 – 54 + 27 + 2 = 2
Step 6: Find absolute extrema
To find the absolute extrema, we also need to evaluate the function at the endpoints of the interval we are interested in. If no specific interval is given, then we consider the limits as x approaches ±∞:
f(x) as x → ±∞: Since the leading term x³ dominates, we have:
- f(x) → ∞ as x → ∞
- f(x) → -∞ as x → -∞
Conclusion
Thus, the relative extrema are:
- Relative maximum at (1, 6)
- Relative minimum at (3, 2)
No absolute maximum exists, while the absolute minimum occurs as the function approaches -∞ at -∞. Therefore, the final summary of the extrema is:
- Relative maximum: (1, 6)
- Relative minimum: (3, 2)
- Absolute minimum: not defined