If I have two local maximums that are same, can I have a global maximum?
I have the following function on the interval $[-1,4]$:
$$f(x) = x^3 - 12x$$
When I graph this function, I see on this closed interval, I have two local/relative maximums, which occur at x=-1 and x=4 and both max out at y=16. My question is can I have a global maximum when the only two local maximums I have are the same? Do both count as global maximums?
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$\begingroup$On the given interval, the function $f(x)=x^3-12x$ has a minimum at $x=2$ since $f'(2)=0$ and $f''(2)>0$. This is the only critical point in the interval, hence the maximum must be at the endpoints. In this case we have that $$f(-1)=11$$ and $$f(4)=16,$$ so that the maximum occurs at $x=4$.
PS. In the case that a function attains its maximum at more than one point (cf. $\sin x$), then you may pick any one of the points which satisfies other constraints of the problem, or if there are none given, list them all.
$\endgroup$ $\begingroup$Since you have a continuous function $f$ on a compact domain $A$ this function is bounded, and the value $\sup_{x\in A}f(x)=:M$ is taken in at least one point of $A$. This value $M$ is then called the (global) maximum of $f$ on $A$, an is denoted by $\max_{x\in A} f(x)$.
In the case at hand the maximal value $16$ is taken at only one point of $[{-1},4]$. As a general rule note that the value $M$ of the maximum is well defined, and changes little under small perturbations of the problem, but the place(s) where this maximum is taken are not a priori uniquely determined and can change abruptly under perturbations.
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