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: Identify the domain of interest.
We need to analyse $f(x) = |x-5| + 2|x-7|$ on the interval $(7, \infty)$.

Step 2: Remove the absolute values for $x > 7$.
When $x > 7$: $x - 5 > 0$ and $x - 7 > 0$. So $|x-5| = x-5$ and $|x-7| = x-7$.

Step 3: Simplify $f(x)$.
\[ f(x) = (x-5) + 2(x-7) = x - 5 + 2x - 14 = 3x - 19. \]

Step 4: Find $f'(x)$.
\[ f'(x) = 3 > 0 \quad \text{for all } x \in (7, \infty). \]

Step 5: Conclude the nature of the function.
Since $f'(x) = 3 > 0$ on $(7, \infty)$, the function is strictly increasing there.

Step 6: State the answer.
$f(x)$ is an increasing function on $(7, \infty)$.
\[ \boxed{\text{Increasing function}} \]