Question

In the interval \[ (7,\infty), \] the function \[ f(x)=|x-5|+2|x-7| \] is:

• A. increasing function
• B. decreasing function
• C. constant function
• D. attains maximum value

Hint:

For modulus functions, first identify the interval and remove the modulus signs according to the sign of each expression inside the modulus.

Answer 1

The Correct Option is A

Step 1: Remove modulus signs in the interval \((7,\infty)\).
For \[ x\gt 7, \] we have \[ x-5\gt 0 \] and \[ x-7\gt 0 \] Therefore, \[ |x-5|=x-5 \] and \[ |x-7|=x-7 \] Thus, \[ f(x)=(x-5)+2(x-7) \] \[ f(x)=x-5+2x-14 \] \[ f(x)=3x-19 \]

Step 2: Determine the nature of the function.
Since \[ f(x)=3x-19 \] is a linear function with positive slope \[ 3\gt 0, \] the function increases as \(x\) increases.
Hence, \[ f(x) \] is an increasing function on \[ (7,\infty) \]

Step 3: Final conclusion.
Therefore, \[ \boxed{\text{increasing function}} \]