I'm learning calculus from a book called "Forgotten Calculus". I'm getting confused by something. The author states that a critical point can be, but need not be, both a relative extreme and an absolute extreme at the same time. Fair enough. But in the case of two simple exercises, I find that either she's not being optimally precise in her wording of the solution, or I'm missing something fundamental.
Consider these two functions:
a) y = f(x) = 1/4x^4 + x and
b) y= f(x) = x^10 + 2
I can find the critical points easily enough. For each there is only one. For a) it's (-1, -3/4) and for b) it's (0, 2).
She says that both are relative minima. Yes, but I think the more complete answer would be to call them absolute minima. My reasoning is that in each case, if the critical point is the ONLY critical point of the function (and having established, via the first derivative test, that each one is a minimum), then by definition each must be an ABSOLUTE minimum. As a check I graphed each function with a TI 83 with the window set quite tall and wide and as far as I can see, indeed, each is an absolute minimum.
Am I right? If not, why not?
I'd be grateful for any advice.