As x ranges over all real values, what is the minimum value of f(x)=|x-123|+|x-456| + |x-789| ?
To find the minimum value of the function f(x) = |x-123| + |x-456| + |x-789| as x ranges over all real values, we can follow these steps:
1. Identify critical points:
- Critical points occur where the absolute value terms change sign. In this case, it happens when (x-123) = 0, (x-456) = 0, and (x-789) = 0. Solve each equation separately to find the critical points.
For (x-123) = 0, x = 123.
For (x-456) = 0, x = 456.
For (x-789) = 0, x = 789.
So, the critical points are x = 123, 456, and 789.
2. Test intervals between critical points:
- Now, we need to evaluate the function f(x) in each interval between the critical points. We will check the endpoints of these intervals as well.
Interval: (-∞, 123]:
Choose test value x = 0 (less than 123) and calculate f(x):
f(x) = |0-123| + |0-456| + |0-789| = 123 + 456 + 789 = 1368
Interval: [123, 456]:
Choose test value x = 300 (between 123 and 456) and calculate f(x):
f(x) = |300-123| + |300-456| + |300-789| = 177 + 156 + 489 = 822
Interval: [456, 789]:
Choose test value x = 600 (between 456 and 789) and calculate f(x):
f(x) = |600-123| + |600-456| + |600-789| = 477 + 144 + 189 = 810
Interval: [789, +∞):
Choose test value x = 1000 (greater than 789) and calculate f(x):
f(x) = |1000-123| + |1000-456| + |1000-789| = 877 + 544 + 211 = 1632
3. Find the minimum value:
- Compare the values obtained in step 2 and choose the minimum one.
The minimum value of f(x) is 810 which occurs in the interval [456, 789].
Therefore, the minimum value of f(x) = |x-123| + |x-456| + |x-789| as x ranges over all real values is 810.