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Chapter 1: Problem 43

Determine the intervals over which the function is increasing, decreasing, or constant. $$ f(x)=|x+1|+|x-1| $$

Short Answer

Expert verified
Therefore, the function \(f(x)=|x+1|+|x-1|\) is decreasing for \(x<-1\), increasing for \(-1鈮鈮�1\), and increasing for \(x>1\).

Step by step solution

01

Break the function into different intervals

The function \(f(x)=|x+1|+|x-1|\) can be written differently depending on whether \(x\) is less than -1, between -1 and 1, or greater than 1. We can express the function as follows: \n\(f(x) = -(x+1) - (x-1)\) for \(x<-1\), \n\(f(x) = (x+1) - (x-1)\) for \(-1 鈮� x 鈮� 1\), and \n\(f(x) = (x+1) + (x-1)\) for \(x>1\).
02

Evaluate the derivative of the function in each interval

For \(x<-1\), \(f鈥�(x)= -2\). For \(-1 鈮� x 鈮� 1\), \(f鈥�(x)= 2\). For \(x>1\), \(f鈥�(x)= 2\).
03

Define the intervals of increase, decrease or constant function

If the derivative of the function is positive, then the function is increasing. If the derivative is negative, the function is decreasing. If the derivative is zero, the function is constant. From the evaluated derivatives: \n The function is decreasing for \(x<-1\) (because the derivative \(f鈥�(x)=-2<0\)), \n The function is increasing for \(-1鈮鈮�1\) (because \(f鈥�(x)=2>0\)), and \n The function is increasing for \(x>1\) (because \(f鈥�(x)=2>0\)).

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