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Question
If a<b<c<d & xϵR then the least value of the function,
f(x)=|x−a|+|x−b|+|x−c|+|x−d| is
If a<b<c<d & xϵR then the least value of the function,
f(x)=|x−a|+|x−b|+|x−c|+|x−d| is
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Solution
The correct option is A c+d−b−a
f(x)=⎧⎪
⎪
⎪
⎪
⎪⎨⎪
⎪
⎪
⎪
⎪⎩(a+b+c+d)−4xx<a(b+c+d)−a−2xa≤x<bc+d−a−bb≤x<c2x−(a+b+c)+dc≤x<d4x−(a+b+c+d)x≥dWe can see that upto b, graph is decreasing and it is constant from b to c and incrasing after c.
Hence, from the graph ,its clear that least value of the function is (c+d−a−b)
f(x)=⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩(a+b+c+d)−4xx<a(b+c+d)−a−2xa≤x<bc+d−a−bb≤x<c2x−(a+b+c)+dc≤x<d4x−(a+b+c+d)x≥d
We can see that upto b, graph is decreasing and it is constant from b to c and incrasing after c.
Hence, from the graph ,its clear that least value of the function is (c+d−a−b)
Hence, from the graph ,its clear that least value of the function is (c+d−a−b)
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