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\textbf{Partial Differentiation}

\textit{\textbf{Functions of more than one variable}}
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\textbf{Question}

For each constant $C$, $f(x,y,z)=C$ is a plane intercepting $C^3$ on
the $x$-axis,
$2C^3$ on the $y$-axis and $3C^3$ on the $z$-axis. 

Find $f(x,y,z)$.


\textbf{Answer}

If the level surface $f(x,y,z)=C$ is the plane
$$\frac{x}{C^3}+\frac{y}{2C^3}+\frac{z}{3C^3}=1$$
that is, $x+\frac{y}{2}+{z}{3}=C^3$, then
$$f(x,y,z)= \left ( x + \frac{y}{2}+ \frac{z}{3} \right )^(1/3).$$

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