What is m if the function f(x) =|x −m+1|−|x −3m+3| is decreasing?

Answer 1

This is the difference of two absolute values. For all values of m except m = +1, the right hand term is greater than the left hand term and so f(x) is decreasing. When x = 1, f(x) = 0 for all real x , so it is not decreasing.

So, m can be any real number, except m = 1

Answer 2

To answer this question, one must consider several cases.

First one must write each absolute value term as a piecewise function.

|x-m+1| = |x-(m-1)| = x-(m-1) if x >= m-1, and m-1-x if x< m-1,

|x-3m+3| = |x-3(m-1)| = x-3(m-1) if x >= 3(m-1), and 3(m-1)-x if x< 3(m-1)

Second on must consider the whether m-1 or 3(m-1) is larger.

If m >= 1, then 3(m-1) >= m-1, otherwise m-1 > 3(m-1),

Then one must break down f(x) into a piecewise function for both cases of m. Note that if m=1, f(x)=0 for all x, which can be considered decreasing, but not strictly decreasing. However all values of m lead to intervals over which f(x) is constant, so no value of m will lead to a strictly decreasing function over all x. m>1 leads to two intervals over which f(x) is constant and one interval over which f(x) is strictly increasing, and m<1 leads to two intervals over which f(x) is constant and one interval over which f(x) is strictly decreasing.

So any value of m<=1 will work.