BIZNIQUE / CALCULUS / 45
Recognise the geometry of a convex function
Recognise the geometry of a convex function
Compare a curve with the straight chord between two points.
Embedded Files
THE IDEA
Read the example.
Read the example.
For a convex function on a convex domain, the graph does not rise above the chord between any two points. A local minimum is therefore global. The example uses a quadratic and moves the point being compared with the chord.
f((1−t)x + ty) ≤ (1−t)f(x) + tf(y)
Use it well
Use it well
Convex does not mean strictly convex, and a global minimum need not be unique or attained. Constraint geometry and the domain also matter.
TRY IT YOURSELF
A quick check.
A quick check.
For f(x) = x², compare f(0) with the chord midpoint between x = −2 and x = 2.
Check your reasoning
Check your reasoning
f(0) = 0, while the chord midpoint has height 4. The convexity inequality holds.
Page updated
Google Sites
Report abuse