Finding Local Maximum and Minimum Points
- Local maximum and minimum points are where a function reaches its highest or lowest value within a small interval.
- These points are also called stationary points because the derivative of the function is zero at these locations.
Steps to Find Local Maximum and Minimum Points
- Find the Derivative: Calculate the first derivative, $f'(x)$.
- Identify Stationary Points: Solve $f'(x) = 0$ to find potential maximum or minimum points.
- This could either lead to the local minimum or the local maximum
- Check by graphing the function which each value corresponds to!
These maxima's and minima's are local which means higher values can exist in the entire domain of the function, but around a certain range from either side of the point, that value is the highest value.
ExampleLet $f(x) = \frac{1}{3}x^3 + 3x^2 + 8x + 1$, then to find the stationary points, you
- Find $f'(x)$ such that you have
$$f'(x) = x^2 + 6x + 8$$
- Then you equate the derivative to 0 and solve
$$x^2 + 6x + 8 = 0 \Longrightarrow x = -2 \text{ OR } x = -4$$
- When you check the graph on your GDC, notice $x=-4$ leads to the maximum while $x=-2$ leads to the minimum
