For our project on Maxima and Minima of functions, we have to do functions of type $\frac{k}{|x-a|+|x-b|}$.

So, I chose $f(x)=\frac{2}{|x-1|+|x-2|}$

I noticed that the derivative is positive for $x<1$, equal to zero for $1 \le x <2$ and negative for $x \ge 2$. So, the derivative does not change its sign at a certain point, but over an interval. In fact the graph of $f(x)$ looks like this-


So, my question is, what is the point of maximum? Because there are infinite points where the function attains maximum value.

  • $\begingroup$ I'm pretty sure you just say they are all maxima. For example, $f(x)$ achieves its global maximum when $x \in [1, 2]$ $\endgroup$ – Hwai-Ray Tung Jul 26 '16 at 6:29

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