Question

Let \( f: \mathbb{R} \to [0, \infty) \) be a continuous function. Then which one of the following is NOT TRUE? \[ \text{There exists } x \in \mathbb{R} \text{ such that } f(x) = \frac{f(0) + f(1)}{2}. \]

• A. There exists \( x \in \mathbb{R} \) such that \( f(x) = \frac{f(0) + f(1)}{2} \)
• B. There exists \( x \in \mathbb{R} \) such that \( f(x) = f(0) \)
• C. There exists \( x \in \mathbb{R} \) such that \( f(x) = \int_0^1 f(t) dt \)
• D. There exists \( x \in \mathbb{R} \) such that \( f(x) = f(1) \)

Hint:

The Intermediate Value Theorem applies when a function is continuous, but averaging integrals does not guarantee an exact value at any specific point.

Answer 1

The Correct Option is D

Step 1: Analyzing the problem.
This problem deals with the properties of continuous functions. By the Intermediate Value Theorem, for any value between \( f(0) \) and \( f(1) \), there exists an \( x \) such that \( f(x) = \frac{f(0) + f(1)}{2} \), and similarly for other options.
Step 2: Analyzing option (C).
We cannot guarantee that there exists an \( x \) such that \( f(x) = f(1) \) because the function could have a value between \( f(0) \) and \( f(1) \) but never reach \( f(1) \), depending on its behavior.
Step 3: Conclusion.
Thus, the correct answer is (D).