# Problem of the Day

If , for what -value is the value of minimized?

A.

B. 1

C. 2

D. 3

E.

### Solution

[latexpage]

First, simplify the expression as follows:

$y=|6x-18|+|-7x+7|+|3x+2|$

$y=6|x-3|+|(-7)(x-1)|+3|x+\frac{2}{3}|$

$y=6|x-3|+7|x-1|+3|x+\frac{2}{3}|$

Each expression within the absolute value bars will be minimized at $x=3,x=1,x=-\frac{2}{3}$, respectively. Examine the rates of change for the expressions as $x$ approaches each of these values. As $x$ approaches $\frac{2}{3}$ from the left, all three expressions are decreasing in value at rates of $-6, -7$, and $-3$ respectively, for a cumulative rate of change of $-16$. For $x$ values greater than $\frac{2}{3}$ and less than $1$, the last expression is increasing at a rate of 3, but the other two are decreasing at a rate of $-6$ and $-7$, with a cumulative rate of change of $-10$. For values of $x$ greater than $1$, the first expression is still decreasing at a rate of $-6$, but the others are increasing at a rate of $7$ and $3$, for a cumulative rate of change of positive $4$. $x=1$ marks the first point where the cumulative rate of change moves from negative to positive, and hence must provide the minimum value for $y$.

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