Draw the graph of `f(x) =y= |x-1|+3|x-2|-5|x-4|` and find the values of `lamda` for which the equation `f(x) = lamda` has roots of opposite sign.

Explanation

To solve the problem step by step, we will first define the function \( f(x) = |x-1| + 3|x-2| - 5|x-4| \) in different intervals based on the critical points \( x = 1, 2, \) and \( 4 \). Then we will graph the function and find the values of \( \lambda \) for which the equation \( f(x) = \lambda \) has roots of opposite signs.

### Step 1: Define the function in different intervals

1. **For \( x < 1 \)**:
- \( |x-1| = 1-x \)
- \( |x-2| = 2-x \)
- \( |x-4| = 4-x \)
...