Home > Calculus > Identifying Turning Points (Local Extrema) for a Function

What is the minimum of #f(x)=|x-1|+|x-2|+cdots+|x-1391|# function?

Answer 1

Emma Aldridge

#483720#

We know that

#sum_(k=1)^(k=1391)abs(x-k) = sum_(k=-695)^(k=695) abs(x-k)#

Now the minimum is located by symmetry, at #x = 0# so

#min f(x) = f(0)=sum_(k=-695)^(k=695) abs(-k) = 2 xx ((696 xx 695)/2) = 695 xx 696 =483720#

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Answer 2

Axel Alvarez

The minimum value of the function ( f(x) = |x-1| + |x-2| + \cdots + |x-1391| ) occurs at the median of the set of numbers ( {1, 2, \ldots, 1391} ), which is ( x = 696 ). Therefore, the minimum value of the function is ( f(696) = 1390 ).

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Answer 3

Isabella Ackerman

The minimum of the function ( f(x) = |x-1| + |x-2| + \ldots + |x-1391| ) occurs when ( x ) is the median of the set ({1, 2, \ldots, 1391}), which is (x = 696). The minimum value of the function is (f(696) = 695).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.