Second Derivative Test
f″ > 0 means concave up; f″ < 0 concave down.
Key idea
f″ > 0 means concave up; f″ < 0 concave down.
In depth
If f′(c) = 0 and f′′(c) > 0, the graph is concave up (cup) and c is a local min; f′′(c) < 0 gives a local max. If f′′(c) = 0 the test is inconclusive.
Worked example
f(x) = x³ − 3x: f′′ = 6x. At x = 1, f′′ = 6 > 0 so a minimum.
Formula
Real-world uses
- Deciding whether a turning point is a maximum or minimum in engineering optimisation
Common mistake
Concluding there is an inflection when f′′ = 0 without checking for a sign change.
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 Second Derivative Test?
f″ > 0 means concave up; f″ < 0 concave down. It is a topic at the expert level in Calculus.
What is the formula for Second Derivative Test?
The key formula is x^3-a*x.
Where is Second Derivative Test used in real life?
Real-world examples: Deciding whether a turning point is a maximum or minimum in engineering optimisation.
What should I learn before Second Derivative Test?
Make sure you are comfortable with First Derivative Test, the previous topic in Calculus.
What is a common mistake with Second Derivative Test?
Concluding there is an inflection when f′′ = 0 without checking for a sign change.
How can I practise Second Derivative Test?
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 Second Derivative Test?
Move on to Inflection Points next.
Related expert topics in Calculus
- Derivatives of Inverse Trig – Derivatives of arcsin, arccos, arctan.
- Implicit Differentiation – Differentiate both sides of F(x,y)=0 treating y as a function of x.
- Logarithmic Differentiation – Take logs before differentiating products and powers.
- Related Rates – Link rates of change of connected quantities.
- Linear Approximation – Approximate f near a using its tangent line.
- Differentials – Use dy = f′(x)dx for small changes.