Past exam of the mathematical methods II course of the University of Oxford / 2013 resit / 6 / Solution

The gradient is $\nabla f=(8x,2y)$, so $\nabla f(1,1)=(8,2)$. The supplied direction is not a unit vector; normalize it to $\hat{\mathbf{u}}=(1,2)/\sqrt{5}$. Its directional derivative isThe direction of greatest decrease is opposite to the gradient:The Cauchy-Schwarz inequality shows that no other unit direction can give a more negative derivative.