Question

Largest value of \( \min(2 + x^2, 6 - 3x) \), when \( x>0 \), is

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

When maximizing a minimum, equalize the two expressions to find the balance point.

Answer 1

The Correct Option is C

We are maximizing the minimum of two expressions: \( f_1 = 2 + x^2 \) and \( f_2 = 6 - 3x \). For the minimum to be as large as possible, set them equal: \[ 2 + x^2 = 6 - 3x \] \[ x^2 + 3x - 4 = 0 \] \[ (x + 4)(x - 1) = 0 \] Positive root: \( x = 1 \). At \( x = 1 \), both expressions are: \[ 2 + 1 = 3, \quad 6 - 3 = 3 \] Hence the maximum possible minimum value is 3.