Question

QC: Consider f(x)=abs(x-3)+abs(x-7) and the point (5,4).

i) Is it an absolute/global min, according to the definition in our textbook? Explain.

ii) Is it a relative/local min, according to the definition in our textbook? Explain.


QD: this is related to the way we measure errors in statistics and machine learning.

i) What x value minimizes the sum of the absolute distances from the values 1, 3, and 8?

ii) Repeat but for sum of squared distances rather than absolute distances.

iii) How do your answers in (i) and (ii) relate to the mean and the median of 1, 3, and 8?

iv) What x value minimizes the sum of the absolute distances from the values 1, 3, 7, 8?

v) What x value minimizes the sum of the squared distances from the values 1, 3, 7, 8?

vi) is your observation in (iii) still true?

Answer 1

you know that the graph will have cusps at x = 3 and x=7 but it will not go clear down to the x-axis because between 3 and 7, both absolute values are positive.

Go to any good online graphing site, such as desmos or wolframalpha and check out the graph. Things should be easier after that.