How do you find critical points on a graph?

Question

Luke Alvarez · Accepted Answer

Critical points for a function #f# are numbers (points) in the domain of a function where the derivative #f'# is either #0# or it fails to exist. So look for places where the tangent line is horizontal (#f'(c)=0#) Alternatively, the tangent line is vertical and there are cusps and discontinuities (jump or removable) where the tangent line does not exist.

Emily Ammerman · Answer

undefined To find critical points on a graph, follow these steps:

1. Determine the derivative of the function.
2. Set the derivative equal to zero and solve for $x$. These points are potential critical points.
3. Determine the second derivative of the function.
4. Evaluate the second derivative at each potential critical point.
5. If the second derivative is positive at a potential critical point, the function has a local minimum at that point.
6. If the second derivative is negative at a potential critical point, the function has a local maximum at that point.
7. If the second derivative is zero or undefined at a potential critical point, further analysis is needed (such as using the first derivative test or higher-order derivative tests).
8. Critical points may also occur at points where the derivative is undefined (such as vertical tangents) or where the function has cusps or corners.

These steps help identify critical points, which are points where the derivative of the function is zero or undefined, and they are essential for analyzing the behavior of functions and determining features such as local extrema and inflection points.