Critical Points and Curve Sketching
Where f′ = 0 or is undefined, the graph may peak or dip.
About this topic
Critical points occur where f′(x) = 0 or is undefined; they are candidates for local maxima and minima. The first derivative test checks whether f′ changes sign, while the second derivative test uses concavity: f″ positive means a minimum and f″ negative means a maximum.
A complete sketch uses a checklist: find the domain and intercepts, locate asymptotes, find critical points and test intervals for increasing or decreasing, then use the second derivative to find concavity and inflection points where f″ changes sign. Plot these key points and join them smoothly.
Example: for f(x) = x³ − 3x, f′(x) = 3x² − 3 = 0 gives x = ±1. Since f″(x) = 6x, f″(1) = 6 is positive, giving a local minimum at (1, −2), and f″(−1) = −6 is negative, giving a local maximum at (−1, 2). The inflection point is at x = 0.
This analysis reveals the shape of data trends in economics, science and design. A critical point is not necessarily an extremum: for x³, f′(0) = 0 but the function keeps increasing. Also remember to check endpoints on closed intervals and to use the original function, not the derivative, to find the y-values.
Key ideas
- f′ > 0 means increasing
- f′ < 0 means decreasing
- The second derivative tells concavity
Quick check
Use cases: where Critical Points and Curve Sketching helps
- Homework: review the key idea behind Critical Points and Curve Sketching before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Critical Points and Curve Sketching without notes.
- Visual learning: play with the interactive lab to see how Critical Points and Curve Sketching changes when you adjust its inputs.
- Teaching a friend: explain Critical Points and Curve Sketching in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Critical Points and Curve Sketching?
Where f′ = 0 or is undefined, the graph may peak or dip.
What will I learn in Critical Points and Curve Sketching?
f′ > 0 means increasing. f′ < 0 means decreasing. The second derivative tells concavity.
Is Critical Points and Curve Sketching part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Critical Points and Curve Sketching lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Critical Points and Curve Sketching?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.