Extreme Values of Functions
of Several Variables

Calculus III Handout

March 17, 2009

Recall that if $S$ is a subset of $n$-dimensional space and $P$ is a point of $S$ we say that $P$ is a point in the interior of $S$ or a point inside $S$ if there is some (small) positive number $r$ such that every point of $n$-dimensional space within distance $r$ of $P$ is a point of $S$.

Recall that a function $f$ of $n$ variables is differentiable at a point inside its domain if it admits first order approximation by a linear function near the given point.

Theorem.   If a function $f$ of $n$ variables has an extreme value for the subset $S$ of its domain at a point $P$ of $S$ that is a point inside the domain of $f$ where $f$ is differentiable, then the gradient vector $\nabla f\left(P\right)$ of $f$ at $P$ must be perpendicular to the tangent vector at $P$ of every differentiably parameterized curve lying in $S$ and passing through $P$.

Proof. Let $G\left(t\right)$ be a differentiably parameterized curve contained in $S$ and passing through $P$ when $t=a$. Since $S$ is contained in the domain of $f$, the function $h\left(t\right)=f\left(G\left(t\right)\right)$ is defined for all values of $t$ for which $G\left(t\right)$ is defined, and since $f$ is differentiable at $P=G\left(a\right)$, the function $h$ is differentiable at $a$. In fact, the “chain rule” tells us that $$h^{\prime }\left(a\right)=\nabla f\left(P\right)·G^{\prime }\left(a\right)\text{.}$$ Since $f$ has an extreme value relative to the set $S$ at the point $P$ and each $G\left(t\right)$ is in $S$, it follows that $h$, a function of one variable, has a local extreme value at $t=a$, and, therefore, that $h^{\prime }\left(a\right)=0$. Consequently, $\nabla f\left(P\right)$ is perpendicular to the tangent vector $G^{\prime }\left(a\right)$ of the curve at $P$.

Corollary 1.   If a function $f$ of $n$ variables has an extreme value for the subset $S$ of its domain at a point $P$ of $S$ that is a point inside $S$ where $f$ is differentiable, then the gradient vector $\nabla f\left(P\right)$ must be the zero vector.

Proof. If $P$ is a point inside $S$ then every sufficiently short line segment passing through $P$ must be perpendicular to $\nabla f\left(P\right)$, which means that every vector must be perpendicular to $\nabla f\left(P\right)$.

Corollary 2.   If a function $f$ of $n$ variables has an extreme value for the subset $S=\left\{g=0\right\}$ of its domain at a point $P$ of $S$ where $f$ and $g$ are differentiable functions, then the gradient $\nabla f\left(P\right)$ of $f$ and the gradient $\nabla g\left(P\right)$ of $g$ must be parallel vectors.

Proof. The statement is formally true, but probably useless if $\nabla g\left(P\right)=0$. We assume that $\nabla g\left(P\right)$ is not the zero vector. In this case $\nabla g$ is perpendicular to the tangent hyperplane (i.e., plane if $n=3$ or line if $n=2$) to $S$ at $P$. Every unit vector in the tangent hyperplane is tangent to some small differentiably parameterized curve segment lying in $S$ and passing through $P$. Hence, by the theorem, $\nabla f\left(P\right)$ is also perpendicular to each such curve segment, and, hence, to the tangent hyperplane. Since a hyperplane has only one parallel class of normal vectors, $\nabla f\left(P\right)$ and $\nabla g\left(P\right)$ must be parallel.

Remark. The theorem is useful also in the case where $f$ is a function of $3$ variables and the constraint set $S$ is a curve in space. Then the fact that $P$ lies in $S$ corresponds roughly to two equations for $P$ and the orthogonality condition of the theorem provides, in non-degenerate situations an additional equation with the result that (usually) only finitely many such $P$ are possible. (Among these are points that are maxima, minima, and those that are neither.) This is equivalent to the principle of “Lagrange multipliers” discussed in the text.