 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2013-resit/6/solution

\nabla f = (8 x, 2 y)

, so

\nabla f (1, 1) = (8, 2)

. The supplied direction is not a unit vector; normalize it to

\hat{u} = (1, 2) / 5

D_{\hat{u}} f = (8, 2) \cdot \frac{( 1 , 2 )}{5} = \frac{12}{5} .

The direction of greatest decrease is opposite to the gradient:

\hat{n}_{decrease} = - \frac{( 8 , 2 )}{68} = - \frac{( 4 , 1 )}{17}, D_{\hat{n}_{decrease}} f = - 217 .

The Cauchy-Schwarz inequality shows that no other unit direction can give a more negative derivative.

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