Tangent Line Trick

Bound a convex summand by its tangent line, then sum.

The idea

The tangent line trick proves inequalities of a standard shape: show that a sum $f(x) + f(y) + f(z)$ is at least some constant, given a linear constraint such as $x + y + z = 3$, with equality expected when the variables are equal.

The method replaces each summand by a linear lower bound and adds. The bound comes from convexity.

Proposition (Tangent line bound).

If $f$ is convex on an interval containing $a$ and $x$, then $f(x) \;\ge\; f(a) + f'(a)\,(x - a).$ That is, a convex function never dips below its tangent line at $a$.

Choose $a$ to be the common value of the variables at the expected equality point. The tangent bound is then exact at that point, so no slack enters where the original inequality is tight.

Summing the bound over the variables, the constant terms accumulate, and the slope terms collect into a single multiple of $x + y + z$, which the constraint fixes to a number. The sum is therefore bounded below by a constant, and the several-variable problem never had to be solved as one.

The method requires $f$ to be convex on the range the variables can occupy. Where $f$ is not convex, the tangent line can cross above the graph, and the bound fails.

Ways to work on it

• Walkthrough. Build the tangent line bound and sum it under a linear constraint.
• Practice. Compute the tangent line of a convex summand at the equality point.
• Hardest. Prove a full cyclic bound by summing tangent lines.

Not sure where to start? Take the ten-question placement test.