To solve this problem, let's use the given conditions step by step.
1. Understand the conditions:
- The function [tex]\( f(x) \)[/tex] is even. This means that for any [tex]\( x \)[/tex], [tex]\( f(-x) = f(x) \)[/tex]. Consequently, if [tex]\( (a, 0) \)[/tex] is an intercept, then [tex]\( (-a, 0) \)[/tex] is also an intercept.
2. Known intercept:
- Given that [tex]\( (6, 0) \)[/tex] is an intercept, it implies that [tex]\( (-6, 0) \)[/tex] must also be an intercept.
3. Total intercepts:
- We need to find the set of points that accounts for all five [tex]\( x \)[/tex]-intercepts. Hence, we should check each option if they together make a plausible set of five intercepts including [tex]\( (6, 0) \)[/tex] and [tex]\( (-6, 0) \)[/tex].
Now, let's evaluate each set of points:
1. First set [tex]\((-6,0), (-2,0), (0,0)\)[/tex]:
- We already have the intercepts [tex]\( (6, 0) \)[/tex] and [tex]\( (-6, 0) \)[/tex].
- Adding [tex]\( (-2, 0) \)[/tex] and [tex]\( (0, 0) \)[/tex] makes it four points: [tex]\( (-6, 0), (-2, 0), (0, 0), \)[/tex] and [tex]\( (6, 0) \)[/tex]. This set still lacks the fifth intercept, so this option is incomplete.
2. Second set [tex]\((-6,0), (-2,0), (4,0)\)[/tex]:
- Again, we start with [tex]\( (6, 0) \)[/tex] and [tex]\( (-6, 0) \)[/tex].
- Adding [tex]\( (-2, 0) \)[/tex] and [tex]\( (4, 0) \)[/tex] makes it four points: [tex]\( (-6, 0), (-2, 0), (4, 0), \)[/tex] and [tex]\( (6, 0) \)[/tex]. This set also lacks the fifth intercept and hence is insufficient.
3. Third set [tex]\((-4,0), (0,0), (2,0)\)[/tex]:
- Consider [tex]\( (6, 0) \)[/tex] and [tex]\( (-6, 0) \)[/tex].
- Including [tex]\( (-4, 0), (0, 0), \)[/tex] and [tex]\( (2, 0) \)[/tex] forms five points: [tex]\( (-6, 0), (-4, 0), (0, 0), (2, 0), \)[/tex] and [tex]\( (6, 0) \)[/tex].
- However, this set does not include [tex]\( (-2, 0) \)[/tex]. So it can be discarded.
4. Fourth set [tex]\((-4,0), (-2,0), (0,0)\)[/tex]:
- Using [tex]\( (6, 0) \)[/tex] and [tex]\( (-6, 0) \)[/tex],
- Adding [tex]\( (-4, 0), (-2, 0), \)[/tex] and [tex]\( (0, 0) \)[/tex] results in five points: [tex]\( (-6, 0), (-4, 0), (-2, 0), (0, 0), \)[/tex] and [tex]\( (6, 0) \)[/tex].
- This list of points is missing one intercept.
5. Fifth set should be examined logically:
- We should look for both symmetry and completeness for five intercepts.
- [tex]\( (-6, 0), (-2, 0), (0, 0), (6, 0) \)[/tex] is mentioned as potentially correct.
- Summarizing these points incorporates the given, [tex]\( (6, 0) \)[/tex] and logically provided correct pairs.
To conclude: the only set that correctly lists all five intercepts is [tex]\((-6, 0), (-2, 0), (0, 0), (6, 0)\)[/tex], making sure each provided criterion and required [tex]\(x\)[/tex]-intercepts fit logically:
[tex]\[
\boxed{(-6, 0), (-2, 0), (0, 0), (6, 0)}
\][/tex].
