For how many integer values of x will f(x) = |x - 3| + |x - 11| + |x -

Question

For how many integer values of x will f(x) = |x - 3| + |x - 11| + |x - 19| + |x - 31| assume a minimum value?

A. 7
B. 9
C. 11
D. 12
E. 13

SatvikVedala · Answer

Hi Bunuel f(x) will have 9 domains x<3,x=3, ..... x>31. each domain has local minima value hence 9 But since x is an integer & considering region between x>11 and x<19 which cancels out x, the answer is 7

Oppenheimer1945 · Answer

At symmetrical values  , the x terms cancels out . Hence for 9 values we obtain min value.

B)

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gmatophobia · Answer

Modulus represents the distance between two points. Hence, |a - b| represents the distance between 'a' and 'b' Let's plot the points on a number line as shown below - 1.png The extreme points are 31 and 3, hence the minima will lie at the mid point of the line segment. Point of Minima = \frac{31 - 3 }{ 2} = 14 2.png Total Distance = 11 + 3 + 5 + 17 = 36 Let's see what happens when we move the right of 14, i.e. at point 15 3.png Total Distance = 12 + 4 + 4 + 16 = 36 We can also deduce this logically. As we move towards the right, the distance between the chosen point (15 in this case) and the points on the right of the chosen point (19 and 31 in this case) reduces by 1 unit, and the distance of the points to its left (3 and 11 in this case) increases by 1. Hence, the increase and the decrease cancel out, and the sum of distance = 36. This continues until the value of x is 19. Post 19, say when the value of x is 20, we have three points to the left of x (i.e. 3, 11, and 19) and only one value on the right of x (i.e. 31). Hence for each movement to the further right, the distance between x and the point to its right (i.e. 31) reduces by 1, while the distance between x and the points to its left (i.e. 3, 11, and 19) increases by 1. Hence, the net increase = 3 - 1 = 2. The same observation can be made when the value of x is less than or equal to 14. Therefore, for each value of x from 11 to 19, both inclusive, the sum of distances will remain constant. Number of points = 19 - 11 + 1 = 9 Option B

bumpbot · Answer

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For how many integer values of x will f(x) = |x - 3| + |x - 11| + |x -

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For how many integer values of x will f(x) = |x - 3| + |x - 11| + |x -

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