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).