The minimum value of |x| + |x + frac{1}{2}| + | vedclass

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(D) Let $f(x) = |x| + |x + \frac{1}{2}| + |x - 3| + |x - \frac{5}{2}|$.This is a sum of absolute value functions. The minimum value of a sum of the fo

Vedclass · Accepted Answer

$6$

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(D) Let $f(x) = |x| + |x + \frac{1}{2}| + |x - 3| + |x - \frac{5}{2}|$.This is a sum of absolute value functions. The minimum value of a sum of the form $\sum |x - a_i|$ occurs when $x$ is the median of the values $a_i$.Here,the critical points are $0, -\frac{1}{2}, 3, \frac{5}{2}$.Arranging these in ascending order: $-\frac{1}{2}, 0, \frac{5}{2}, 3$.The median lies in the interval $[0, \frac{5}{2}]$.For any $x$ in the interval $[0, \frac{5}{2}]$,let's evaluate $f(x)$:$f(x) = x + (x + \frac{1}{2}) + (3 - x) + (\frac{5}{2} - x) = x + x + \frac{1}{2} + 3 - x + \frac{5}{2} - x = \frac{1}{2} + 3 + \frac{5}{2} = 6$.Thus,for any $x \in [0, \frac{5}{2}]$,the value of the function is constant and equal to $6$.Therefore,the minimum value is $6$.

The minimum value of $|x| + |x + \frac{1}{2}| + |x - 3| + |x - \frac{5}{2}|$ is

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