Answer
We begin by calculating the level curve that corresponds to point $P$ by replacing into $f(x, y) \rightarrow f(1,2) .$ In this problem, algebra works fine.
\[
\begin{array}{c}
3-2(2)+4(1)=f(1,2) \\
3-2 y+4 x=3 \\
-2 y+4 x=0 \\
2 x=y
\end{array}
\]
We next invoke the gradient formula $f_{x} \mathrm{i}+f_{y}=\nabla f \mathbf{j}$ to determine the gradient.
\[
\begin{array}{c}
-2=f_{y} \text { and } 4= f_{x}\\
4 \mathbf{i}-2 \mathbf{j}=\nabla f
\end{array}
\]
Work Step by Step
We begin by calculating the level curve that corresponds to point $P$ by replacing into $f(x, y) \rightarrow f(1,2) .$ In this problem, algebra works fine.
\[
\begin{array}{c}
3-2(2)+4(1)=f(1,2) \\
3-2 y+4 x=3 \\
-2 y+4 x=0 \\
2 x=y
\end{array}
\]
We next invoke the gradient formula $f_{x} \mathrm{i}+f_{y}=\nabla f \mathbf{j}$ to determine the gradient.
\[
\begin{array}{c}
-2=f_{y} \text { and } 4= f_{x}\\
4 \mathbf{i}-2 \mathbf{j}=\nabla f
\end{array}
\]
