Question

A function f(x) is defined on the interval with values in R. It satisfies \( \int_0^2 f(x)[x-f(x)]dx = \frac{2}{3} \). Find the value of f(1).}

Hint:

When an integral equation involves quadratic terms of an unknown function, try to arrange it into the integral of a perfect square set to zero. This is a common trick that leads to a direct solution for the function.

Answer 1

Step 1: Understanding the Question:
We are given an integral equation involving an unknown function \(f(x)\) and we need to determine the value of this function at a specific point, \(x=1\).
Step 2: Key Insight and Approach:
The expression within the integral, \(xf(x) - [f(x)]^2\), looks like part of a perfect square. Let's try to construct a perfect square involving these terms. Consider the expression \((k \cdot x - f(x))^2\) for some constant \(k\). \[ (kx - f(x))^2 = k^2x^2 - 2kxf(x) + [f(x)]^2 \] This doesn't quite match. Let's try rearranging the given integral equation first. \[ \int_0^2 (x f(x) - [f(x)]^2) dx = \frac{2}{3} \] Consider the integral of a non-negative function. Let's analyze the expression \(\int_0^2 \left( \frac{x}{2} - f(x) \right)^2 dx\). \[ \int_0^2 \left( \frac{x}{2} - f(x) \right)^2 dx = \int_0^2 \left( \frac{x^2}{4} - x f(x) + [f(x)]^2 \right) dx \] We can split the integral: \[ = \int_0^2 \frac{x^2}{4} dx - \int_0^2 (x f(x) - [f(x)]^2) dx \] Step 3: Detailed Calculation:
We know the value of the second integral from the problem statement. Let's calculate the first integral: \[ \int_0^2 \frac{x^2}{4} dx = \left[ \frac{x^3}{12} \right]_0^2 = \frac{2^3}{12} - \frac{0^3}{12} = \frac{8}{12} = \frac{2}{3} \] Now substitute the values back into our expression: \[ \int_0^2 \left( \frac{x}{2} - f(x) \right)^2 dx = \frac{2}{3} - \left( \frac{2}{3} \right) = 0 \] The integrand, \( \left( \frac{x}{2} - f(x) \right)^2 \), is a square, so it is always non-negative. The only way the integral of a non-negative continuous function over an interval can be zero is if the function itself is zero everywhere in that interval. Therefore, we must have: \[ \left( \frac{x}{2} - f(x) \right)^2 = 0 \quad \text{for all } x \in \] \[ \frac{x}{2} - f(x) = 0 \] \[ f(x) = \frac{x}{2} \] Step 4: Final Answer:
We have found the function \(f(x) = x/2\). Now we can find the value of \(f(1)\): \[ f(1) = \frac{1}{2} = 0.5 \]