If $z$ is a complex number and the minimum value of $|z| + |z - 1| + |2z - 3|$ is $\lambda$ and if $y = 2[x] + 3 = 3[x - \lambda]$ then find the value of $\frac{1}{5}([x + y])$. (where $[.]$ denotes the greatest integer function).
Step-by-Step Solution
Key Concept: The minimum value $\lambda = 2$ is found by analyzing the piecewise function, then use the constraint equation to determine $[x]$, calculate $y$, and apply the greatest integer function.
To find the minimum of $|z| + |z-1| + |2z-3|$, let $z = x$ (real). We minimize $f(x) = |x| + |x-1| + |2x-3|$. Critical points occur at $x = 0, 1, 3/2$. Evaluating: $f(0) = 0 + 1 + 3 = 4$, $f(1) = 1 + 0 + 1 = 2$, $f(3/2) = 3/2 + 1/2 + 0 = 2$. For $x \in [1, 3/2]$, $f(x) = x + (1-x) + (3-2x) = 4 - 2x$ decreases from 2 to 2, so $\lambda = 2$. Given $2[x] + 3 = 3[x-2]$, let $[x] = n$, so $2n + 3 = 3(n-2) = 3n - 6$, giving $n = 9$. Thus $[x] = 9$ means $x \in [9, 10)$. Then $y = 2(9) + 3 = 21$, so $x + y \in [30, 31)$ and $[x+y] = 30$. Therefore $\frac{1}{5}[x+y] = \frac{30}{5} = 6$.
Correct Answer: 6