Past exam of the mathematical methods II course of the University of Oxford / 2021 resit / 5 / Solution

The gradient of the scalar field isFor a unit vector $\hat{\mathbf{n}}$, the directional derivative is $D_{\hat{\mathbf{n}}}f=\nabla f\cdot \hat{\mathbf{n}}$. The Cauchy-Schwarz inequality bounds its magnitude by $\Vert \nabla f\Vert =2\sqrt{3041}$, with equality precisely in the two directions parallel to the gradient. ThereforeThe plus choice gives the greatest positive directional derivative, and the minus choice gives the most negative directional derivative. Both maximize its magnitude.