A property is called global if it relates to the entire object of interest.
For example, the function \(f(x)\) has a global minimum at \(x_0\) if \(f(x_0)\le f(x)\) for all values of \(x\) in the domain of the function.
Here, \(f(x)=x^3-3x\) does not have a global minimum at \(x=1\) because there are values of \(x\) with \(f(x)<f(1)\) (for example, \(f(-3)=-18\) while \(f(1)=-2\)). However, the function does have a local minimum at \(x=1\).
If the domain of this function were restricted to \(x\ge0\), then the function would have a global minimum at \(x=1\).
A global minimum can occur at the end of the domain, even if this is not a stationary point, as in this sketch of a function with domain \([-2,2]\), which has its global minimum at \(x=2\):
A global maximum is defined similarly.
