Thread: Surface defined on the domain?

"Let S be the surface z = e^(x+y) - 2, defined on the domain D = {(x,y) ∈ ℝ^2 | x ≥ 0 and y ≥ 0}. What is the minimal value of z for points on S?"

With this question, I thought that since both the x- and y-values need to be greater than zero for the domain, then obtaining the minimal value would simply mean subbing 0 into the equation. I did that, and got the answer, -1. (Which also matches up as being the correct answer to the question). So, I just basically want to know was what I did to get the answer correct or is there some other way you get the answer? (As, I may be wrong but it just seems that it was a bit of a straightforward and simple approach with getting the answer.)

what you did is correct. it can be explained as follows:
1) the value of z completely depends on the value of x+y.
2)the lower the value of x+y the lower the value of z.
3) the least value of x+y available in D is 0.
so e^0-2 is the answer.

Here the situation was fairly simple so we could use the above reasoning. In general such problems are solved using multivariate calculus.

We have



The second inequality is true for all so, is a lower bound for



On the other hand we get that bound: .


Edited: Sorry, Ididn't see the previous post.