ATAR Notes: Forum
VCE Stuff => VCE Mathematics => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Mathematical Methods CAS => Topic started by: Samueliscool223 on August 09, 2021, 11:15:12 am
-
So I know you can use f''(x) to determine this, but it doesnt work all the time, e.g. for the function f(x)=x^3: f"(x) implies there is a point of inflection, and the function is concave down/up on the left/right when in actuality there is only a stationary point of inflection and no local max or min. for cubics, only the family of functions f(x)=x^3-ax would satisfy this property, but the 2nd derivative test would (erroneously) suggest otherwise. so clearly this method can be crap sometimes, and so i ask, is there a superior method to the f''(x) one (excluding methods involving the use of tech obviously)??
-
Trial and error?
-
I'm not sure what you mean by it doesn't work all the time. A continuous curve doesn't have to have a local min/max for it to be concave up/down, The second derivative will tell you this concavity without fail provided the function is twice differentiable across the domain you wish to analyse. In the example you suggest, for \(x^3, x > 0\), the curve is concave up. There can be no arguments about that. I'm more curious as to where you got this idea from because it's not something I think would've been said explicitly.
If you want a better visualiser, think of a more primitive definition of concave. You should be able to draw a straight line between any two points on the curve and enclose some area under the curve. There doesn't necessarily have to be a turning point for this to happen.
The only case where I'd agree that the second derivative test is bad is if the derivative of a function is difficult to find, though realistically this won't happen (as such the second derivative is the most common way of doing so. If you already have a stationary point, a sign table for the first derivative is also a handy option). I will never agree to it being logically incorrect in the way you suggest (if I understand you correctly, apologies if I haven't).
Hope this helps :)