Question

Function, \(f(x) = -|x-1|+5, \forall x \in R\) attains maximum value at x =

• A. 1
• B. 5
• C. 2
• D. 9

Hint:

For functions of the form \(k|x-a|+c\), the vertex (and the extremum) is always at \(x=a\). If \(k\) is negative, it's a maximum. If \(k\) is positive, it's a minimum. The value of the extremum is \(c\). Here, \(k=-1\), so it's a maximum at \(x=1\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The function involves an absolute value term. The maximum or minimum value of such a function can often be found by analyzing the absolute value expression.

Step 2: Key Formula or Approach:
1. The absolute value expression \(|x-a|\) is always non-negative, i.e., \(|x-a| \geq 0\). 2. The minimum value of \(|x-a|\) is 0, and this occurs at \(x=a\). 3. We can use this property to find the maximum value of the given function \(f(x)\).

Step 3: Detailed Explanation:
The given function is \(f(x) = -|x-1| + 5\).
Consider the term \(|x-1|\). By definition of absolute value, its value is always greater than or equal to zero. \[ |x-1| \geq 0 \] The minimum value of \(|x-1|\) is 0, which occurs when \(x-1 = 0\), or \(x=1\).
Now consider the term \(-|x-1|\). Multiplying an inequality by -1 reverses the inequality sign: \[ -|x-1| \leq 0 \] This means the maximum value of the term \(-|x-1|\) is 0. This maximum occurs when \(|x-1|\) is at its minimum, which is at \(x=1\).
The function \(f(x)\) is simply \(-|x-1|\) shifted up by 5. Therefore, the maximum value of \(f(x)\) will also occur at the same x-value.
The maximum value of \(f(x)\) is \((\text{max value of } -|x-1|) + 5 = 0 + 5 = 5\).
This maximum value is attained at \(x=1\).

Step 4: Final Answer:
The function \(f(x)\) attains its maximum value at \(x=1\).