Convex Functions and Sets
Functions whose chords lie above the graph.
Key idea
Functions whose chords lie above the graph.
In depth
A set is convex if it contains the line segment between any two of its points; a function is convex if f(λx + (1 − λ)y) ≤ λf(x) + (1 − λ)f(y). Every local minimum of a convex function is global, making optimisation tractable.
Worked example
f(x) = x² is convex (f′′ = 2 > 0); sin x is not.
Formula
Real-world uses
- Guaranteeing that a cost minimisation has a single best answer in economics and data science
Common mistake
Assuming any bowl-looking function is convex without checking.
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 Convex Functions and Sets?
Functions whose chords lie above the graph. It is a topic at the expert level in Differential Equations, Abstract Algebra & Applied Math.
What is the formula for Convex Functions and Sets?
The key formula is f(λx+(1−λ)y) ≤ λf(x)+(1−λ)f(y).
Where is Convex Functions and Sets used in real life?
Real-world examples: Guaranteeing that a cost minimisation has a single best answer in economics and data science.
What should I learn before Convex Functions and Sets?
Make sure you are comfortable with Gradient Descent, the previous topic in Differential Equations, Abstract Algebra & Applied Math.
What is a common mistake with Convex Functions and Sets?
Assuming any bowl-looking function is convex without checking.
How can I practise Convex Functions and Sets?
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 Convex Functions and Sets?
Move on to Linear Programming and Simplex next.
Related expert topics in Differential Equations, Abstract Algebra & Applied Math
- Exact Equations – M dx + N dy = 0 with Mᵧ = Nₓ.
- Bernoulli Equations – Substitute v = y¹⁻ⁿ to linearise.
- Runge–Kutta Methods – Higher-accuracy numerical solvers.
- Second-Order Homogeneous Equations – Use the characteristic equation.
- Undetermined Coefficients – Guess the particular solution form.
- Variation of Parameters – General method for particular solutions.