decide if 3 variables function has a minimum and maximum value

Question

There is a question in my textbook that says decide if the function

$$f(x,y,z) = 2x^3+2y^3+2z^3-3xy-3yz-3zx$$

has a minimum and maximum value on R^3

And it says that the solution is to look at the limits

$$\lim_{t\to\infty} f(t,t,t) = \lim_{t\to\infty} (6t^3-9t^2) = \infty$$ And do the same but with $$-\infty$$

My question is why would you look at the function when x=y=z ? I guess that is what the solution example shows?

Answer 1

has a minimum and maximum value on $\mathbb{R}^3$

If they mean a global minimum or maximum, then the example from the solution shows that the function does not have those: the function values tend to $+\infty$ and $-\infty$ respectively, for $(t,t,t)$ with $t\to\pm\infty$. They chose to look at these points, because it supports their claim.

You don't have to look at points of the form $(t,t,t)$, as long as you find points that support the argument you're trying to make. You could e.g. also look at $f(t,0,0)=2t^3$ and draw the same conclusion - with less effort I would even say!

Note that the function does have a local minimum (of $-3$ at $(1,1,1)$).

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As an illustration, another example where you could use the same approach: $f(x,y)=x^2-y^2$ has no minimum or maximum on $\mathbb{R}^2$ since its values become:

• arbitrarily large for points of the form $(x,0)$ where you take $x$ sufficiently large;
• arbitrarily small for points of the form $(0,y)$ where you take $y$ sufficiently large.

Answer 2

You don't need to look at that, you could choose also $x=t$ and $y=z=0$. It is just one option which give you an answer.

$$\lim_{t\to\infty} f(t,0,0) = \lim_{t\to\infty} 2t^3 = \infty$$

$$\lim_{t\to-\infty} f(t,0,0) = \lim_{t\to-\infty} 2t^3 = -\infty$$