Question

The minimum value of \( f(x) = \max\{x, 1+x, 2-x\} \) is

• A. \( \frac{1}{2} \)
• B. \( \frac{3}{2} \)
• C. \( 1 \)
• D. \( 0 \)
• E. \( 2 \)

Hint:

For max-min problems, equate expressions to find balance point.

Answer 1

The Correct Option is B

Concept: Minimum of max function occurs when all expressions are as equal as possible.

Step 1: Identify expressions.
\[ x,\quad 1+x,\quad 2-x \]

Step 2: Compare \( x \) and \( 2-x \).
\[ x = 2-x \Rightarrow x=1 \]

Step 3: Evaluate function at \( x=1 \).
\[ f(1) = \max\{1,2,1\} = 2 \]

Step 4: Try balancing other pairs.
\[ 1+x = 2-x \Rightarrow x = \frac{1}{2} \]

Step 5: Evaluate at \( x=\frac{1}{2} \).
\[ f = \max\left(\frac{1}{2}, \frac{3}{2}, \frac{3}{2}\right) = \frac{3}{2} \] Minimum value: \[ \frac{3}{2} \]