Relative Extrema
By Sebastian Wright
I did a question to find relative extrema for the following function: $f(x)=x^2$ on $[−2,2].$
The answer said that there is no relative maxima for this function because relative extrema cannot occur at the end points of a domain. Why is this so ? Thank you.
$\endgroup$2 Answers
$\begingroup$There is a relative minimum at $0$. But there are also relative maxima at $2$ and at $-2$.
The definition for a relative maximum at a point $x_0$ is that $f(x)\leq f(x_0)$ for all $x$ in a sufficiently small neighborhood of $x_0$ (intersected with the function's domain).
For $x_0=2$, for example, it is true that $f(x)\leq f(2)$ for all $x$ in the sufficiently small intervall $(1.5,2.5)\cap[-2,2]=(1.5,2]$ around the point $x_0=2$. So, it is a relative maximum.
There is no reason to believe that relative extrema cannot occur at the boundaries of intervals.
$\endgroup$ 12 $\begingroup$The answer is correct. Why it is so follows immediately from the definition.
There's a difference between an extreme value and a relative extreme value of a function at a point. While your function has a maximum value at $\pm2,$ it does not have a relative maximum at all. The reason is that relative extreme values by definition can only be defined at a point $c$ first, when the function is defined over an interval of the form $[c-\delta,c+\delta],$ where $\delta>0.$ Another example is $\sqrt x,$ which has no relative minimum at $x=0,$ though it has a minimum there.
In mathematics, the words are very important and never superfluous. Pay attention to such innocuous qualifiers as relative.
Good luck in your studies.
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