[Mailbag] What Do You Do with the Ideas You Used to Call “Mistakes”

Guillaume ParĂ©, in the very interesting comments of my last post where I
urged us to reconsider mistakes: I do agree with what is written, but I am
still w...
1 week ago
The answer is c = f((a+b)/2).
ReplyDeleteSince f is continuous and increasing, it has an inverse function g. By turning the graph sideways, we see that the area A(c) is given by the following equation.
A(c) = int (f(a) to c) [g(x)  a] dx + int (c to f(b)) [b  g(x)] dx.
Using the fundamental theorem of calculus,
A'(c) = 2g(c)  a  b,
so the minimum area occurs when A'(c) = 0, or when g(c) = (a+b)/2, which is equivalent to c = f((a+b)/2).