Past exam of the mathematical methods II course of the University of Oxford / 2014 / 3 / Solution

The gradient is $\nabla f=(2x+y,x)$ and at $(1,1)$ it is $(3,1)$. For a unit vector $\hat{\mathbf{n}}$, the directional derivative is $D_{\hat{\mathbf{n}}}f=(3,1)\cdot \hat{\mathbf{n}}$. The Cauchy-Schwarz inequality bounds its absolute value by $\sqrt{10}$, with equality in the parallel or antiparallel direction. ThusThe plus sign gives greatest increase; the minus sign gives greatest decrease, with the same magnitude. Stationary directions at the specified point have zero first derivative, so they are perpendicular to the gradient:These are tangents to the local level curve, not a claim that $(1,1)$ is itself a stationary point of the two-variable function.