I was sitting here, thinking after taking a test on areas between two functions, about a problem that I cannot find an answer to online, and as well, may be outright silly, but I figured I would propose it here anyway, as I am truly stuck.
Find f(x) such that the area of the shape bounded by it, x=0, x=1, y=0, y=1 and f'(x) is a maximum.
If I were not new here, I would post an example, but alas, I cannot.
Now, for this problem, I would assume that f(0)=0, f(1)=1, f'(0)=0, and f'(1)=1, in order to keep the shape closed about the tiny section of graph that is open to it. Obviously, this question could be expanded to fit larger sections of the graph instead of the tiny box bounded by all of the vertical and horizontal lines. But I am curious as to how one would go about a question like this, as I am genuinely unsure, as a student in Calculus II. Thank you for your help! And maybe, if the solution is found, a general solution could be found for when f(b)=b and f'(b)=b (A.K.A. f(b)=f'(b)).