- #1
anemone
Gold Member
MHB
POTW Director
- 3,883
- 115
Determine the minimum of $y$ where $y=|x-1|+|2x-1|+|3x-1|+\cdots+|nx-1|$.
Bacterius said:The minimum of $y$ is zero and is achieved when $n = 1$ and $x = 1$.
Seriously though, should we minimize $y$ over $x$ as a function of $n$? Also, are $x, y$ real? (I assume $n$ is a natural number).
For $x>1$: $y=x-1+2x-1+3x-1+4x-1=10x-4$ | |
For $\dfrac{1}{2}<x<1$: $\begin{align*}y&=-(x-1)+2x-1+3x-1+4x-1\\&=-x+2x+3x+4x+1-1-1-1\\&=8x-2\end{align*}$ | For $\dfrac{1}{3}<x<\dfrac{1}{2}$: $\begin{align*}y&=-(x-1)-(2x-1)+3x-1+4x-1\\&=-x-2x+3x+4x+1+1-1-1\\&=4x\end{align*}$ |
For $\dfrac{1}{4}<x<\dfrac{1}{3}$: $\begin{align*}y&=-(x-1)-(2x-1)-(3x-1)+4x-1\\&=-x-2x-3x+4x+1+1+1-1\\&=-2x+2\end{align*}$ | For $x<\dfrac{1}{4}$: $\begin{align*}y&=-(x-1)-(2x-1)-(3x-1)-(4x-1)\\&=-x-2x-3x-4x+1+1+1+1\\&=-10x+4\end{align*}$ |
An absolute value function is a mathematical function that returns the distance between a number and zero on a number line. It is represented by two vertical bars surrounding the input, and always outputs a non-negative value.
An absolute value function is written as |x|, where x is the input value. It can also be written as abs(x) or ||x||.
The domain of an absolute value function is all real numbers, as any number can be used as the input. The range of an absolute value function is also all real numbers, but the output will always be non-negative.
No, an absolute value function will always return a non-negative value. However, the input value can be negative, which will result in a positive output.
An absolute value function is used in many real life situations, such as calculating distances, determining the magnitude of a vector, and finding the difference between two values. It is also used in various mathematical models and equations in fields such as physics and economics.