Say I have a function $f\left(x\right)=\left|\frac{x^{2}-2}{x^{2}-1}\right|$. I have to find the local minima/maxima of the function. The point is that this function is non-differentiable at $x=\pm\sqrt{2},\pm1$. When i try to break free of the modulus, I end up getting a piece-wise function. How do I find the extrema points in this case ( I do not want to go by graph method) ? Because, sure I can find the critical points, which in this case is $x=\pm\sqrt{2},\pm1,\ 0$. But how do I use the first derivative test to check whether these points are actually extrema or not.
So basically my question is how do we go about finding local minima/maxima without any graphical method?
Any hints or approaches are welcome.
I don't want answer of this particular question, I just want to know the concept/method.
Best Answer
Final Answer: (to summarize)
In general, if we have a function of the form f(x)=|g(x)|, then you can note that f(x)⩾0 for all x in the domain. Moreover, if there are points at which g(x) is zero, then they must be global minima for the function f. (USEFUL POINT TO REMEMBER)
If you know a function f:U→R for U open has a singularity at a point then limx→p|f(x)|=∞, This means that it is for certain that f has either no global min or no global max.
If yet there is any other critical point, just use the fundamental definition that $f\left(c^{-}\right)>f\left(c\right)\ and\ f\left(c^{+}\right)>f\left(c\right)$ for minima and like that we can do for maxima. This can be used in a piece-wise function too at the point where the definition changes, to find if it is an extrema or not.
It is not a hard and fast method and depending upon the questions, things can change, nothing can be said exactly. Best way to solve these questions is the graphical approach wherever applicable.