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Question 5 : The minimum value of f(x)=|3-x|+|2+x|+|5-x| is equal to __________.
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Explanatory Answer
The distance between two points of a number line is the difference between the larger point and the smaller point.
If the value of the two points p and q are unknown, then the distance is modulus of their difference.
|p - q| = |p - q|
We can re name the points as |x - 3|, |x -(- 2)|, |x - 5|
The points -2 , 3 , 5 need to be marked on the number line.
Any point we place on the number line, we calculate the distance from that point to the other points -2,3,5.
The minimum value will lie between -2 and 5,because if that point lies outside this range then certain parts between -2 and 5 overlap while calculating the total distance. Hence to be of minimum distance the point has to lie between -2 and 5
Any point between -2 and 5 their sum of distances will add up to 7 .
The extra distance will be between the point and 3 which will be minimum when x = 3, because |3 - x| will go to zero
Hence |x - 3| + |x + 2|+ |5 - x|
The minimum distance 0 + 2 + 5 = 7
The question is " The minimum value of f(x)=|3-x|+|2+x|+|5-x| is equal to __________. "
Hence, the answer is 7
