4.2 Stationary points

We say that $P=(a,b)$ is a stationary point if

where the subscript indicates that we evaluate the partial derivatives at $P$.

Proof. Suppose that $(a,b)$ is a local maximum or a local minimum. Clearly the function $f(x,b)$ has a local maximum or a local minimum. It follows by frame 4.16 of MATH101 that $f(x,b)$ has a stationary point as a function of $x$, so the first–order partial derivative of $f$ with respect to $x$ must be zero at $(a,b)$. Likewise, the partial derivative of $f$ with respect to $y$ is zero at $(a,b)$.