Let $f ( x , y ) = x ^ { 2 } + 3 y ^ { 2 } - 3 x$ Find the gradient vector of f at the point (-1, 2).
A) $\nabla f = - 5 \vec { i } - 12 \vec { j }$
B) $\nabla f = - 5 \vec { i } + 24 \vec { j }$
C) $\nabla f = 5 \vec { i } + 12 \vec { j }$
D) $\nabla f = - 5 \vec { i } + 12 \vec { j }$

Question

Quizplus · Accepted Answer

The gradient vector of $f(x, y)$ is given by $\nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right)$. For $f(x, y) = x^2 + 3y^2 - 3x$, $\frac{\partial f}{\partial x} = 2x - 3$ and $\frac{\partial f}{\partial y} = 6y$. Substituting $x = -1$ and $y = 2$ gives $\nabla f = (-5, 12)$, which corresponds to $-5\vec{i} + 12\vec{j}$.

Let f(x,y)=x2+3y2−3xf ( x , y ) = x ^ { 2 } + 3 y ^ { 2 } - 3 xf(x,y)=x2+3y2−3x

Let $f ( x , y ) = x ^ { 2 } + 3 y ^ { 2 } - 3 x$