Math101 · Independent Ontario learning library
Math101Critical Points
Critical numbers are domain inputs where a derivative is zero or undefined and are candidates for extrema or other changes in behaviour.
Critical numbers tell us where to investigate; they do not automatically prove that a maximum or minimum occurs.
Definition
A number $c$ in the domain of $f$ is a critical number when
or $f'(c)$ does not exist. The corresponding point $(c,f(c))$ is a critical point on the graph.
The input must belong to the original function's domain. A vertical asymptote is not a critical point because no point of the function exists there.
A systematic method
- Determine the domain of $f$.
- Compute $f'(x)$.
- Solve $f'(x)=0$.
- Find domain inputs where $f'$ is undefined.
- Keep only values belonging to the domain of $f$.
- Evaluate or classify the candidates as required.
Worked example
Further testing is needed to classify them.
Undefined derivative example
For $f(x)=|x|$, the derivative is $-1$ for $x<0$ and $1$ for $x>0$, but it does not exist at $x=0$. Since $0$ is in the function's domain, it is a critical number and corresponds to a local minimum.
This is why solving only $f'(x)=0$ can miss important points.
Domain exclusion example
For $f(x)=1/x$, the derivative $f'(x)=-1/x^2$ is undefined at $x=0$. However, $0$ is not in the domain of $f$, so it is not a critical number.
It is a domain boundary and vertical asymptote, which still matters for interval analysis but receives a different label.
Common mistakes
Calling every derivative zero a maximum or minimum. It is only a candidate.
Ignoring points where the derivative is undefined. Keep them if the function exists there.
Including an asymptote as a critical point. The input is not in the domain.
Forgetting interval endpoints in absolute-extrema problems. Compare them separately.
Reporting only input values when points are requested. Evaluate $f(c)$ too.
