Question

Let \( f: [0, 2] \to \mathbb{R} \) be such that \( |f(x) - f(y)| \leq |x - y|^{4/3} \) for all \( x, y \in [0, 2] \). If \( \int_0^2 f(x) \, dx = \frac{2}{3} \), then

 

Hint:

For sums involving smooth functions, you can often approximate the sum by multiplying the integral of the function by the number of terms.

Answer 1

Correct Answer: 673

Step 1: Use the given condition.
The condition \( |f(x) - f(y)| \leq |x - y|^{4/3} \) implies that \( f(x) \) is a very smooth function, and this smoothness suggests that \( f(x) \) is continuous on the interval \([0, 2]\). Hence, the values of \( f \left( \frac{1}{k} \right) \) for large \( k \) should be very close to each other.
Step 2: Estimate the sum.
The sum \( \sum_{k=1}^{2019} f \left( \frac{1}{k} \right) \) can be approximated by evaluating the function at a few points. Since \( f(x) \) is continuous and smooth, and given that the integral of \( f(x) \) over the interval \([0, 2]\) is \( \frac{2}{3} \), we estimate the sum as follows: \[ \sum_{k=1}^{2019} f \left( \frac{1}{k} \right) \approx 2019 \times \frac{2}{3} = 1346. \] Final Answer: \[ \boxed{1346}. \]