Question

relative extrema of polynomials quick check

Answer 1

To quickly find the relative extrema of a polynomial function \( f(x) \), follow these steps:

1. **Find the Derivative**: Compute the first derivative \( f'(x) \).

2. **Set the Derivative to Zero**: Solve the equation \( f'(x) = 0 \) to find critical points.

3. **Determine the Nature of Critical Points**:
- Use the **First Derivative Test**:
- Choose test points in the intervals created by the critical points.
- Observe the sign of \( f'(x) \):
- If \( f'(x) \) changes from positive to negative at a critical point, there is a **relative maximum**.
- If \( f'(x) \) changes from negative to positive, there is a **relative minimum**.
- If there is no change in signs, the critical point is neither a maximum nor a minimum.

- Alternatively, use the **Second Derivative Test**:
- Compute the second derivative \( f''(x) \).
- Evaluate \( f''(x) \) at each critical point:
- If \( f''(x) > 0 \), there is a **relative minimum**.
- If \( f''(x) < 0 \), there is a **relative maximum**.
- If \( f''(x) = 0\), the test is inconclusive.

4. **Evaluate at Critical Points**: Optionally, plug in the critical points back into the original function \( f(x) \) to find the actual values of the relative extrema.

### Example

For a given polynomial \( f(x) = x^3 - 3x^2 + 4 \):

1. **Find the Derivative**:
\[
f'(x) = 3x^2 - 6x
\]

2. **Set the Derivative to Zero**:
\[
3x^2 - 6x = 0 \implies 3x(x - 2) = 0 \implies x = 0, x = 2
\]

3. **Second Derivative**:
\[
f''(x) = 6x - 6
\]

4. **Nature of Critical Points**:
- At \( x = 0 \):
\[
f''(0) = 6(0) - 6 = -6 \quad (\text{relative maximum})
\]
- At \( x = 2 \):
\[
f''(2) = 6(2) - 6 = 6 \quad (\text{relative minimum})
\]

5. **Evaluate**:
- \( f(0) = 4 \) (relative maximum)
- \( f(2) = -2 \) (relative minimum)

So, the relative extrema are at \( (0, 4) \) (maximum) and \( (2, -2) \) (minimum).