6A. Critical Points & First Derivatives
Instructions:
- Use the input box to define the function y = f(x). Use the input boxes for a and b to define the endpoints of the domain of f. Use the checkboxes to include/exclude the endpoints.
- Use the checkbox for Critical Points or Interior Extrema to show/hide critical points or interior extreme values, respectively.
- Use the checkbox for Monotonicity to highlight the section of the graph corresponding to the location of the point c.
- Use the Derivative checkbox to show the graph of the derivative function.
Critical Points & Monotonicity
Our goal is to be able to describe transitions in the behavior of a function (e.g., from increasing to decreasing, from concave up to concave down). These changes in a function correspond to easier-to-find changes in its derivative(s). Because the monotonicity (increasing/decreasing behavior) of a function is tied to the sign (positive/negative) of its derivative, describing the monotonicity of f requires finding where the derivative f' is positive, negative, or zero.
Assuming certain nice properties of the derivative function (i.e., that it is continuous), the derivative could not change from positive to negative (or vice versa) without passing through a value of 0. A critical point is a point where the derivative is equal to zero (or does not exist). Hence, a critical point is a location where the function has the potential to change monotonicity.
The illustration above demonstrates that the critical points of a function give potential candidates for extreme values (maximum/minimum values). While all extreme values occur at critical points, not every critical point results in an extreme value. So, we have to test critical points to determine whether there is a maximum or minimum value there.