Find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes, and one vertex in the plane x+7y+9z=63
Let P(a,b,c) denote the vertex in the plane which produces the largest volume.
1. $\displaystyle V(a,b,c) = a \cdot b \cdot c~\wedge~a+7b+9c=63~\implies~c=\frac19 (63-a-7b)$
Therefore
$\displaystyle V(a,b)=7ab - \frac19 a^2 b -\frac79 ab^2$
2. Differentiate wrt a and wrt b. Both derivatives must be zero:
$\displaystyle \left|\begin{array}{l}7b-\frac29ab - \frac79 b^2=0 \\ \\ 7a - \frac19a^2 - \frac{14}9 ab=0 \end{array} \right.$
3. Solve this system of equations for (a, b). Determine c and afterwards the volume of the box.
4. I've got $\displaystyle V_{max} = 147$