Julio says, "If you subtract 13 from my number and multiply the difference by -3, the result is -54 ." What is Julios's num
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Here’s the answer and each step I took to solve it. With the example I made on the side just continue to repeat that for each one to get the answers. You just replace that one number for the X to take the place for the next X value as you go down the table solving them.
Lagrange multipliers:
(if 
)
(if )
(if 
)
In the first octant, we assume
, so we can ignore the caveats above. Now,
so that the only critical point in the region of interest is (1, 2, 2), for which we get a maximum value of
.
We also need to check the boundary of the region, i.e. the intersection of
with the three coordinate axes. But in each case, we would end up setting at least one of the variables to 0, which would force 
, so the point we found is the only extremum.
In the first octant, we assume
so that the only critical point in the region of interest is (1, 2, 2), for which we get a maximum value of
We also need to check the boundary of the region, i.e. the intersection of