The title of your question and the question itself are asking two very different questions.
When we ask "What condition guarantee that a function has global extremes in $[a,b]$?" we are asking for conditions that are sufficient: if the condition is met, then the function will definitely have global extremes, but it is possible for a function to have global extremes without meeting the conditions. This can also be phrased as saying "which conditions ensure that a function has global extremes?".
However, when we ask "Which condition must be true in order for a function to have global extremes?" we are asking for conditions that are necessary; these are conditions that every function that has global extremes must satisfy, though it is possible for a function to satisfy the conditions and yet not have global extremes.
So the title of your question asks one thing. The quoted text asks something different.
To further complicate things, the final option in your quoted text does not match the question quoted, because the final option again talks about "guarantees" rather than "requirements".
If the question was meant to ask about guarantees (that is, conditions that, if met, will ensure that the function has global extremes), then the correct answers are (1) and (3); (1) guarantees the existence of global extremes by the Extreme Value Theorem. Condition (3) guarantees the existence of global extremes because for all $x$ in $[a,b]$, we have $a\leq x\leq b$, so if $f$ is increasing on the interval this implies $f(a)\leq f(x)\leq f(b)$; so $f(a)$ is a global minimum and $f(b)$ is a global maximum. Condition (2) does not ensure the existence of global extremes. For example, the following function defined on $[0,1]$ is positive at all points, but has neither a maximum nor a minimum: $f(x) = \left\{\begin{array}{ll} 10 &\mbox{if $x=0$;}\\ \frac{1}{x} &\mbox{if $0\lt x\lt\frac{1}{2}$;}\\ 10 &\mbox{if $x=\frac{1}{2}$;}\\ \frac{1}{1-x} &\mbox{if $\frac{1}{2}\lt x\lt 1$;}\\ 10&\mbox{if $x=1$.} \end{array}\right.$ And (4) is incorrect because conditions (1) and (3) are certainly guarantees for the existence of global extremes.
On the other hand, if the question is quoted correctly and is asking about conditions that are necessary (requirements) for the existence of a global maximum, and we rephrase (4) to read:
4 . None of the above conditions are required for the existence of global extremes.
then that would be the correct answer. Continuity and positivity (conditions 1 and 2) are not required; for example, the function $f(x)$ defined on $[0,2]$ $f(x) = \left\{\begin{array}{ll} 2x-1&\mbox{if $0\leq x\leq 1$;}\\ 2x-3&\mbox{if $1\lt x\leq 2$,} \end{array}\right.$ has global minimum $-1$ at $0$, and has global maximum $1$ at both $x=1$ and $x=2$. The function is neither continuous nor always positive. The function is also not increasing over the interval, so this example also shows that (3) is not a requirement.
Finally, note that a constant function does have global extremes. $M$ is the global maximum for $f(x)$ on the set $A$ if and only if:
- There is a point $p\in A$ such that $f(p)=M$; and
- For all points $q\in A$, $f(q)\leq M$.
If $f(x)=k$ for all $x$, then $k$ is a global maximum; similarly for "global minimum."