Question

Let \( \varphi: \mathbb{R} \to \mathbb{R} \) be a differentiable function such that \( \varphi'(1) = 0 \). Let \( \alpha \) and \( \beta \) denote the minimum and maximum values of \( \varphi(x) \) on the interval [2, 3], respectively. Then which one of the following is TRUE?

• A. \( \beta = \varphi(2.5) \)
• B. \( \alpha = \varphi(2.5) \)
• C. \( \beta = \varphi(3) \)
• D. \( \alpha = \varphi(3) \)

Hint:

Always analyze the behavior of the function and its critical points when determining maximum and minimum values.

Answer 1

The Correct Option is C

Step 1: Given conditions.
We are told that \( \varphi'(1) = 0 \), meaning \( x = 1 \) is a critical point of \( \varphi \). The values of \( \varphi(x) \) on the interval [2, 3] are of interest, and we know \( \varphi \) is differentiable.

Step 2: Analyze the given options.
- (A) \( \beta = \varphi(3) \): Correct. Since \( \varphi(x) \) is differentiable and \( \varphi'(1) = 0 \), we can infer that the maximum value of \( \varphi(x) \) occurs at the endpoint \( x = 3 \). Therefore, \( \beta = \varphi(3) \).
- (B) \( \alpha = \varphi(2.5) \): This is incorrect. The minimum value \( \alpha \) does not necessarily occur at \( x = 2.5 \).
- (C) \( \beta = \varphi(2.5) \): This is incorrect. The maximum value \( \beta \) does not necessarily occur at \( x = 2.5 \).
- (D) \( \alpha = \varphi(3) \): This is incorrect. The minimum value \( \alpha \) does not necessarily occur at \( x = 3 \).

Step 3: Conclusion.
The correct answer is (C) as \( \beta = \varphi(3) \), based on the behavior of the function on the interval [2, 3].