What does the notation $\,\max_{t \le\ x}(t^3 - 3t)\,$ mean, and how do I graph it for $x\leq2$?

Question

This was a part of a bigger problem that I was solving. I had to graph the following definition for $x\le2$. $$\max_{t \le\ x}(t^3 - 3t)$$ I do not understand how to graph this. What does this notation mean?

Answer 1

The given expression is a function of $x$. Denote it by $f(x) = \max_{t\leq x} (t^3-3t)$. Now solve this maximization in the usual manner (derivatives and such). Notice: the maximization is with respect to $t$ (so you have $g(t) = t^3-3t$, and you're finding maximum of $g$ in the interval $(-\infty, x]$) and that $x$ appears as the upper bound on the interval so you have to take that into consideration when finding the maximum.

Answer 2

Function

$$g(x)=\max_{x \le t}f(t)$$

should be interpreted as "the best value obtained so far" ("so far" means here "till time $x$")... a way of thinking very familiar to sportmen/women.

Therefore, as the graph of $f$ is increasing on $(-\infty,-1)$, on all this interval, the best performance so far is the present performance. But, on the segment $(-1,1)$, the graph of $f$ "delves". The "best performance so far" is the value of the last "peak", i.e. $f(-1)=2$. Past "time" $x=1$, the performance becomes anew above value $2$ (one begins to "break his own record(s)"), which means that one has again to stick to the present value of $x$.

Passing from $f$ to $g$ is sometimes called the "rising sun" transformation : imagine (in this case, it is no longer a sport mataphor, but a geographical analogy) the sun at $-\infty$ on the $x$ axis lighting the graph of function $f$) as considered here, with horizontal rays over the valleys... You can think as well to a flood analogy: all valleys have been flooded and transformed into lakes...

Conclusion: $g(x)=f(x)$ everywhere but for $x \in (-1,1)$ where its value stays at $f(-1)=2$.

Remark : "best value so far" is also familiar to computer scientists : it is the usual way to get - at the end - a global maximum when parsing a list of numbers.