Solve the problem.
-Find the derivative of the function $f ( x , y ) = x ^ { 2 } + x y + y ^ { 2 }$ at the point $( 3,4 )$ in the direction in which the function decreases most rapidly.
A) $- 3 \sqrt { 26 }$
B) $- \sqrt { 221 }$
C) $- \sqrt { 173 }$
D) $- \sqrt { 167 }$

Question

Quizplus · Accepted Answer

The gradient of the function at the point (3,4) is $\nabla f(3,4) = (6,11)$. The direction in which the function decreases most rapidly is opposite to the direction of the gradient, which is $(-6,-11)$. The derivative of the function in this direction is $\nabla f(3,4) \cdot (-6,-11) = -6(6) - 11(11) = -121 = -\sqrt{221}$.

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