Turn tangent information into global bounds

Try

The tangent line to f(x)=x² at x=2 is 4x−4. Subtract it from the function: x²−(4x−4)=(x−2)²≥0 for every real x. This tangent is a global lower bound. If a tangent has zero slope, its contact point must therefore minimize the function globally. The useful feature is not the appearance of a bowl in a plot, but an inequality valid at every point in the domain.

Predict

Predict whether a positive second derivative at one point proves global convexity. It does not; curvature must be controlled over the whole relevant convex domain. Predict also whether strict convexity guarantees an attained minimum. The strictly convex function e^x on R has infimum zero but never attains it. Convexity supplies global comparison properties, while existence still requires a separate argument such as compactness or coercivity.

Experiment

Compare three functions through curvature and uniqueness. Compute the derivatives yourself before accepting the classification.

Function on R	Convexity property	Minimizer behavior
x²	Strongly convex, μ=2	Unique minimum at 0
x⁴	Strictly convex, not globally strongly convex	Unique minimum at 0
e^x	Strictly convex	No attained minimum

For x⁴, the second derivative is 12x² and vanishes at zero, so no positive uniform lower curvature bound works. Strict convexity can still be proved, for example from the strictly increasing first derivative. Distinguish a nonnegative Hessian criterion from the stronger claim that every Hessian is positive definite.

Understand

On a convex domain, f is convex when f((1−t)x+ty)≤(1−t)f(x)+tf(y). For a differentiable function on an open convex domain, this is equivalent to the first-order inequality f(y)≥f(x)+∇f(x)·(y−x). To derive it from convexity, apply the segment inequality at small positive t, rearrange and take a directional derivative. Conversely, applying the tangent inequality at an interior segment point to both endpoints and taking a weighted sum recovers convexity.

For a C² function, restricting to every line shows convexity is equivalent to a positive-semidefinite Hessian everywhere on an open convex domain. Strong convexity with parameter μ>0 strengthens the tangent bound by adding (μ/2)||y−x||². Under C² smoothness it corresponds to Hessian eigenvalues at least μ. Strict convexity merely makes interior segment inequalities strict for distinct endpoints. It gives at most one minimizer, since two distinct minimizers would have a lower-valued midpoint, but it does not alone provide existence or a uniform curvature constant.

Apply

Nonsmooth convex functions also have supporting bounds. A subgradient g at x satisfies f(y)≥f(x)+g·(y−x) for every y. At zero for f(x)=|x|, testing positive y requires g≤1 and testing negative y requires g≥−1. Thus every g in [−1,1] is a subgradient. In particular zero is a subgradient, directly certifying the global minimum without differentiability.

For f(x,y)=x²+4y², the Hessian eigenvalues are 2 and 8, so μ=2 is the largest Euclidean strong-convexity constant. Change units by replacing y with 100 times another coordinate and observe how the Hessian spectrum changes. Numerical curvature constants are coordinate-dependent; the underlying decision problem must retain its units and scaling interpretation.

Continue

When using convexity, state the domain and the exact inequality doing the work. A graph or a finite Hessian grid is not a proof over a continuous domain. Continue to First- and second-order conditions to separate necessary stationary conditions from sufficient local tests and from the stronger global certificates available when convexity has been established.