Global Extrema

From Nearby Comparisons to Comparisons on the Whole Domain

A local extremum compares a function value with nearby values. A global extremum compares it with every value on the domain. A local minimum, for example, may be lower than all nearby values but still be higher than a value far away. To establish a global conclusion, we must account for the entire domain, not just the behavior around one point.

A global extremum is attained: the relevant value must equal \(f(a)\) for some point \(a\) in the domain. This is different from a supremum or infimum, which need not be a value of the function. Also, every global maximum or minimum is a local maximum or minimum under the relative-neighborhood definition, but the reverse need not hold.

A continuous function on a closed, bounded interval does attain both a global maximum and a global minimum, by the Extreme Value Theorem. Differentiation can then help locate them. The theorem “Candidates for Absolute Extrema on a Closed Interval,” established in “Fermat’s Theorem,” says that when a function is continuous on \([a,b]\) and differentiable on \((a,b)\), its global extrema occur among the endpoints and the interior stationary points. That result identifies where to look; it does not say that every candidate is an extremum. We still compare their function values.

Comparing Candidates on a Closed Interval

The practical method is to list both endpoints and all interior points where the derivative is zero, evaluate the function at each, and compare the resulting values. If the function is not differentiable at some interior point, that point must also be checked directly: the candidate theorem’s differentiability hypothesis does not allow us to omit it. For functions with infinitely many stationary points, the candidate list may require additional analysis rather than a finite table of values.

A common error is to find one stationary point and declare it a global extremum without checking the rest of the domain. Fermat’s Theorem gives a necessary condition for a differentiable interior local extremum, not a guarantee of global optimality. Endpoints can be global extrema even though Fermat’s Theorem does not apply there, and a nondifferentiable point may also be a global extremum.

Derivative Signs Can Prove Global Optimality

Candidate comparison is not the only route to a global conclusion. If the derivative has a consistent sign on each side of a point, the Mean Value Theorem compares the value at that point with every other value in the interval. The following result makes the needed hypotheses explicit, including differentiability on the two sides of the proposed minimum.

Proof. Let \(x\in[a,b]\) with \(x<c\). The function is continuous on \([x,c]\): this follows from its continuity on \([a,b]\). It is differentiable on \((x,c)\), since \((x,c)\subseteq(a,c)\). By the Mean Value Theorem, there is a point \(\xi\in(x,c)\) such that

Here \(f'(\xi)<0\) and \(c-x>0\), so \(f(c)-f(x)<0\). Thus \(f(c)<f(x)\). This argument includes \(x=a\); the Mean Value Theorem only requires differentiability in the open interval \((a,c)\), not at \(a\).

Now let \(x\in[c,b]\) with \(x>c\). The function is continuous on \([c,x]\) and differentiable on \((c,x)\). The Mean Value Theorem gives a \(\eta\in(c,x)\) such that

Since \(f'(\eta)>0\) and \(x-c>0\), we have \(f(x)>f(c)\). This argument also includes \(x=b\). We have proved \(f(x)>f(c)\) for every \(x\in[a,b]\) with \(x\ne c\). Therefore \(c\) is a global minimum, and no other point can be one. \(\square\)

Strict Convexity and Uniqueness

Strict convexity provides a different way to establish uniqueness. Its defining inequality compares the function at a point between two inputs with the corresponding weighted average of their function values. If two distinct points were both global minima, strict convexity would force a still smaller value between them, which is impossible.

Proof. Suppose \(f\) is strictly convex and attains its global minimum at two distinct points \(u,v\in I\). Write the minimum value as \(m\), so \(f(u)=f(v)=m\), and \(f(x)\geq m\) for all \(x\in I\). The midpoint \(w=(u+v)/2\) lies in \(I\), because \(I\) is an interval. Strict convexity, applied to the distinct points \(u,v\) with weight \(1/2\), gives

This contradicts the fact that \(m\) is a global minimum on \(I\). Thus there cannot be two distinct global-minimum points.

For the maximum statement, suppose \(f\) is strictly concave and has a global maximum \(M\) at distinct points \(u,v\). Strict concavity gives

contradicting the definition of \(M\). Therefore the global maximum is unique. \(\square\)

This theorem guarantees uniqueness only when an extremum is attained; strict convexity alone does not ensure that a global minimum exists on an arbitrary interval. On a closed, bounded interval, continuity supplies existence by the Extreme Value Theorem, while strict convexity then supplies uniqueness. The supporting-line inequality and derivative characterizations of convexity developed earlier can help verify convexity, but existence and uniqueness are separate questions.

When a Global Extremum Is Not Attained

Continuity by itself does not guarantee global extrema on every domain. The closed, bounded interval condition in the Extreme Value Theorem matters. For example, define \(r(x)=x\) on the open interval \((0,1)\). For any \(x\in(0,1)\), the point \(x/2\) also belongs to \((0,1)\) and satisfies \(r(x/2)<r(x)\). Likewise, \((x+1)/2\in(0,1)\) and \(r((x+1)/2)>r(x)\). Thus \(r\) has neither a global minimum nor a global maximum on this domain, even though it is continuous and differentiable. Its infimum is \(0\) and its supremum is \(1\), but neither is attained.

The main checks are therefore distinct: first ask whether the domain and hypotheses ensure that an extremum exists; then identify possible locations; finally compare values or use a global argument such as a derivative sign pattern or strict convexity. A local test can help classify behavior near a candidate, but global optimality always concerns the whole domain.