Showing the graph of the functions without showing that there are only two possible integers to work with (since we're looking for solutions in ℕ, not ℝ) on the

Showing the graph of the functions without showing that there are only two possible integers to work with (since we're looking for solutions in ℕ, not ℝ) on the left side of the maximum feels like missing the most important part of the drawing things out. You can even leave the y axis unlabeled, but label the x-axis (especially since we've already determined f(1)=0) and draw some dots on the curve to show that those are the only values that can be in our solution space. "There's a maximum at e, so the only two possible values on the left of e that can be used in these pairs are 1, and 2. But really there's only one, because 1 is the power identity, so we can't use that. If there is a solution, it has to use 2." Which means we're solving one function for one unknown, in the natural numbers. 2ⁿ=n² gives n=4. We have found the only pair of integers that satisfies our constraint function.