Critical Points
Points where f′ = 0 or does not exist.
Key idea
Points where f′ = 0 or does not exist.
In depth
A critical point is where f′(x) = 0 or f′ is undefined. Local maxima and minima can only occur at critical points (in the interior). Finding them is the first step in optimisation.
Worked example
f(x) = x³ − 3x: f′ = 3x² − 3 = 0 → x = ±1.
Real-world uses
- Finding the best design dimensions where a cost or a stress is at its lowest or highest
Common mistake
Assuming every critical point is a max or min. x³ has one at 0 but no extreme.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Critical Points?
Points where f′ = 0 or does not exist. It is a topic at the advanced level in Calculus.
Where is Critical Points used in real life?
Real-world examples: Finding the best design dimensions where a cost or a stress is at its lowest or highest.
What should I learn before Critical Points?
Make sure you are comfortable with Differentials, the previous topic in Calculus.
What is a common mistake with Critical Points?
Assuming every critical point is a max or min. x³ has one at 0 but no extreme.
How can I practise Critical Points?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Critical Points?
Move on to Increasing and Decreasing Functions next.
Related advanced topics in Calculus
- Idea of a Limit – The value a function approaches as the input approaches a point.
- One-Sided Limits – Left and right limits must agree for the limit to exist.
- Limit Laws – Limits of sums, products and quotients are computed from the parts.
- Factoring and Rationalizing Limits – Remove 0/0 indeterminacy with algebra.
- Limits at Infinity – End behaviour of functions.
- Infinite Limits and Asymptotes – Function values grow without bound near a vertical asymptote.