Question

Let max {a, b} denote the maximum of two real numbers a and b. Which of the following statement(s) is/are TRUE about the function \( f(x) = \text{max}\{3 - x, x - 1\}? \)

• A. It is continuous on its domain.
• B. It has a local minimum at \( x = 2 \).
• C. It has a local maximum at \( x = 2 \).
• D. It is differentiable on its domain.

Hint:

The maximum of two linear functions is continuous but not necessarily differentiable at the point where the two functions intersect.

Answer 1

The Correct Option is A, B

The function \( f(x) = \text{max}\{3 - x, x - 1\} \) involves two linear functions, \( 3 - x \) and \( x - 1 \), and the value of \( f(x) \) is the maximum of these two expressions for any given value of \( x \). Let's analyze the statements:
- (A) It is continuous on its domain.
- Since both \( 3 - x \) and \( x - 1 \) are continuous functions, and the maximum of two continuous functions is also continuous, \( f(x) \) is continuous for all real values of \( x \). Hence, this statement is true.
- (B) It has a local minimum at \( x = 2 \). - At \( x = 2 \), the two functions \( 3 - x \) and \( x - 1 \) intersect, and this point is a local minimum for \( f(x) \). Therefore, this statement is true.
- (C) It has a local maximum at \( x = 2 \). - The function does not have a local maximum at \( x = 2 \), as the function transitions from one linear segment to another at this point. Thus, this statement is false.
- (D) It is differentiable on its domain.
- The function \( f(x) \) is not differentiable at \( x = 2 \) because there is a "corner" at this point (a non-smooth transition between the two linear segments). Hence, this statement is false.
Final Answer: (A) It is continuous on its domain.
(B) It has a local minimum at \( x = 2 \).