When the Hessian Does Not Decide
At a critical point, the second derivative test uses the Hessian to describe the function's local quadratic behavior. If the Hessian is semidefinite and singular, that test may not determine whether the point is a local minimum, a local maximum, or neither. Higher-order terms can then distinguish possibilities that the quadratic terms do not.
A useful way to inspect those terms is to restrict the function to simple paths through the point. Along a straight line, the function becomes a one-variable function, and its first nonzero term shows whether the function rises or falls along that line. Coordinate paths are especially easy to calculate, although checking only a few paths cannot in general settle the classification.
In this tutorial, the point under examination is the origin. A translation of coordinates gives the same analysis at any point \(\mathbf{a}\): use displacements \((u,v,w)=(x-a_1,y-a_2,z-a_3)\) and compare \(f(\mathbf{a}+(u,v,w))\) with \(f(\mathbf{a})\).
Leading Terms Along Lines
Suppose \(f\) has continuous partial derivatives through order \(m\) near the origin. Its Taylor expansion there consists of homogeneous polynomials of degrees \(0,1,\ldots,m\), together with a remainder that is \(o(\|\mathbf{h}\|^m)\). Here \(\mathbf{h}=(x,y,z)\), and \(o(\|\mathbf{h}\|^m)\) means that the remainder divided by \(\|\mathbf{h}\|^m\) tends to zero as \(\mathbf{h}\) tends to zero.
At a critical point, the degree-one term vanishes. If the first nonzero Taylor term after the constant term has degree \(m\), write it as \(P_m(x,y,z)\). It is homogeneous: \(P_m(t\mathbf{v})=t^mP_m(\mathbf{v})\). Thus the expansion takes the form
For a fixed direction \(\mathbf{v}\ne\mathbf{0}\), substitute \(\mathbf{h}=t\mathbf{v}\). The homogeneity of \(P_m\) gives the one-variable expansion
When \(P_m(\mathbf{v})\ne 0\), this leading term determines the sign of the difference for sufficiently small nonzero \(t\), subject to the parity of \(m\). If \(m\) is odd, the sign changes when \(t\) changes sign. If \(m\) is even, the sign along that line is determined by \(P_m(\mathbf{v})\) for either direction of travel.
A Sign Criterion and Why It Works
The leading homogeneous term can sometimes establish an extremum, not just provide evidence along individual lines. For this conclusion, it must have a strict sign in every direction.
For the minimum case, the unit sphere is compact and \(P_m\) is continuous. It therefore has a positive minimum \(c\) on the unit sphere. For any nonzero \(\mathbf{h}\), homogeneity gives \(P_m(\mathbf{h})\geq c\|\mathbf{h}\|^m\). The remainder divided by \(\|\mathbf{h}\|^m\) tends to zero, so for all sufficiently small \(\mathbf{h}\) its absolute value is less than \((c/2)\|\mathbf{h}\|^m\). Consequently,
This proves a strict local minimum. Applying the same argument to \(-f\) proves the maximum case. The uniform bound over all directions is essential: knowing that the leading term is positive on a few tested lines does not provide this bound.
Worked Example: A Strict Minimum from a Positive Leading Term
Worked Example: A Strict Minimum from a Positive Leading Term
Classify the origin for \(f(x,y,z)=x^4+y^4+z^4\). First, \(f(\mathbf{0})=0\), and
All three partial derivatives vanish at the origin. The second partial derivatives are \(f_{xx}=12x^2\), \(f_{yy}=12y^2\), \(f_{zz}=12z^2\), and the mixed partial derivatives are zero. Hence the Hessian at the origin is the zero matrix, so the second derivative test does not decide the classification.
The first nonzero term is the degree-four polynomial \(P_4=x^4+y^4+z^4\). For any nonzero \((x,y,z)\), at least one coordinate is nonzero, and its fourth power is positive; the other fourth powers are nonnegative. Thus \(P_4>0\) in every nonzero direction.
For a direct line check, let \(\mathbf{v}=(a,b,c)\ne\mathbf{0}\). Then
The coefficient \(a^4+b^4+c^4\) is positive. For example, with \(\mathbf{v}=(1,-2,2)\), it equals \(1+16+16=33\), so \(f(t,-2t,2t)=33t^4>0\) whenever \(t\ne0\). More generally, the expression is positive for every nonzero point, not only along the displayed line. Therefore the origin is a strict local minimum—in fact, \(f(x,y,z)>f(\mathbf{0})\) for every nonzero point.
As a check, \(f\) is a sum of nonnegative fourth powers and vanishes only when \(x=y=z=0\). This agrees with the classification.
Coordinate Paths Can Reveal a Saddle
The three coordinate paths through the origin are \((t,0,0)\), \((0,t,0)\), and \((0,0,t)\). They are easy to restrict to, but they sample only three directions. A mixed term can be invisible on all three axes and still change the sign between them.
Worked Example: The Coordinate Axes Miss the Saddle Direction
Consider \(f(x,y,z)=x^4+y^4+z^4-3x^2y^2\). The first partial derivatives are
They all vanish at the origin. Every second partial derivative also vanishes there, so the Hessian is the zero matrix. The first nonzero term is the degree-four polynomial \(P_4=f\) itself.
On the coordinate axes, the restrictions are
All three restrictions are positive for \(t\ne0\). But now use the line \((t,t,0)\). Substitution gives
Thus \(f(t,0,0)>f(\mathbf{0})=0\), while \(f(t,t,0)<f(\mathbf{0})\) for every \(t\ne0\). Both paths approach the origin as \(t\) tends to zero. Values both above and below \(f(\mathbf{0})\) occur arbitrarily close to the origin, so the origin is a saddle point.
The check is direct: the positive axis value is \(t^4\), and the diagonal value is \(-t^4\); these have opposite signs for every nonzero \(t\). The coordinate-axis checks alone would not have found the saddle.
Odd-Degree Terms and Opposite Directions
An odd-degree leading term has a useful feature: reversing a direction reverses its sign. This often makes a saddle easy to detect, even when the quadratic terms vanish. The function's behavior along other paths may involve higher powers, so each restriction should be computed separately.
Worked Example: A Cubic Term Forces Saddle Behavior
Let \(f(x,y,z)=x^3+y^4+z^6\). Its first partial derivatives are
They vanish at the origin. All second partial derivatives vanish there as well, so the Hessian is zero. On the \(x\)-axis, the restriction is \(f(t,0,0)=t^3\). For \(t>0\) this is positive; for \(t<0\) it is negative.
The other two coordinate restrictions illustrate why different paths may have different leading orders:
These two restrictions are positive for nonzero \(t\), but the \(x\)-axis restriction takes both signs arbitrarily near the origin. Therefore the origin is a saddle point. As a check, the points \((t,0,0)\) and \((-t,0,0)\), for \(t>0\), have function values \(t^3\) and \(-t^3\), respectively.
A Common Mistake: Treating Selected Paths as a Proof
A frequent error is to check a few paths, observe that the function increases along each, and conclude that the point is a minimum. The calculation in the preceding saddle example shows why this inference fails. For \(x^4+y^4+z^4-3x^2y^2\), the coordinate-axis restrictions are all \(t^4\). The incorrect conclusion would be:
The implication is not valid: those equalities establish behavior only on the three axes. The diagonal restriction gives \(f(t,t,0)=-t^4<0\), while the \(x\)-axis restriction gives \(f(t,0,0)=t^4>0\). The corrected conclusion is that the origin is a saddle.
The same caution applies when checking many lines or a finite collection of other paths. Positive values along those selected paths are not, by themselves, a proof of positivity in a full neighborhood. A local-extremum conclusion from a leading term requires control in every direction, such as the strict positive or negative condition in the theorem.
A Practical Path-Restriction Procedure
For an inconclusive degenerate critical point, use paths as a way to find decisive sign information. The method can prove a saddle as soon as it finds both signs. A minimum or maximum requires a stronger argument that covers all sufficiently nearby points.
Write the function in displacement coordinates so that the point being classified is the origin.
Use the local expansion to identify nonzero higher-order terms after the constant and vanishing linear terms.
Substitute a line such as \((ta,tb,tc)\), or a coordinate path, and calculate the leading nonzero term of the resulting one-variable function.
Opposite signs on paths prove a saddle. A one-sided sign on selected paths is only evidence; to prove an extremum, establish the sign in a whole neighborhood.
Why Higher-Order Path Analysis Matters
The Hessian records only quadratic behavior. When its quadratic form is semidefinite and singular, some directions may have no quadratic change at all, and terms of degree three or higher can control what happens. Restricting to paths makes these terms concrete: a polynomial in several variables becomes an expression in one parameter, where its leading power and coefficient can be inspected.
The method is particularly effective for finding saddle points, because two paths with opposite signs are conclusive. It also supports a strict-extremum proof when the first nonzero homogeneous term has a strict sign in every direction. Between these situations lies an important limitation: a collection of paths may leave unexamined directions, so a same-sign result along selected paths does not settle the classification.
Check Your Understanding
For each question, distinguish what a path calculation proves from what it merely suggests.
- For \(f(x,y,z)=x^4+y^4+z^4\), what is the leading term along the line \((t,-t,2t)\), and what does its sign imply?
- For a degree-four leading term \(P_4\), suppose \(P_4(\mathbf{v})>0\) in one direction and \(P_4(\mathbf{w})<0\) in another. What classification follows, assuming the expansion remainder is of smaller order?
- Why do positive restrictions on the three coordinate axes fail to prove that the origin is a local minimum?
- If the first nonzero term has odd degree and is nonzero in some direction, what happens to its sign when that line is traversed in the opposite direction?
- What additional condition on a leading homogeneous term allows it to establish a strict local minimum rather than merely positive values along selected lines?