Question

Find all the continuous positive functions \(f(x),\) for \(0 \leq x \leq\) such that \(\int_{0}^{1} f(x) d x=1, \int_{0}^{1} f(x) x d x=\alpha,\) and \(\int_{0}^{1} f(x) x^{2} d x=\alpha^{2}\) where \(\alpha\) is a real number

Short answer

The continuous function \(f(x)\), under the given conditions, would take the form \(f(x)=a+bx+cx^{2}\), with the constants \(a\), \(b\), and \(c\) found through a system of equations. After obtaining these constants, the derived function would satisfy all given conditions.

Step-by-step solution

State Provided Information

Given that \(f(x)\) is a continuous positive function in the interval \([0,1]\) and it satisfies the following equalities: \(\int_{0}^{1} f(x) d x=1, \(\int_{0}^{1} f(x) x d x=\alpha,\) and \(\int_{0}^{1} f(x) x^{2} d x=\alpha^{2}\) where \(\alpha\) is a real number.

Identify the form of the function

Based on the provided conditions and because f(x) is a continuous positive function, the function \(f(x)\) could be a polynomial of degree 2 such that the integral of \(f(x)\) over the domain \([0,1]\) meets the given conditions. Let's assume the function \(f(x)\) in the form \(f(x)=a+bx+cx^{2}\). The constants \(a\), \(b\), and \(c\) are to be determined.

Using the first condition to find the value of \(a\)

Let's use the first condition \( \int_{0}^{1} f(x) d x=1 \) To find \(a\): \(\int_{0}^{1} (a+bx+cx^{2}) d x=1 \). Solving this will give \( a+\frac{b}{2}+\frac{c}{3}=1\)

Using the second condition to find the value of \(b\)

Let's use the second condition \(\int_{0}^{1} f(x) x d x=\alpha \). To find \(b\): \( \int_{0}^{1} (ax+bx^{2}+cx^{3}) d x=\alpha \). Solving this will give \( \frac{a}{2}+\frac{b}{3}+\frac{c}{4}=\alpha \)

Using the third condition to find the value of \(c\)

Let's use the third condition \(\int_{0}^{1} f(x) x^{2} d x=\alpha^{2} \). To find \(c\): \( \int_{0}^{1} (ax^{2}+bx^{3}+cx^{4}) d x=\alpha^{2} \). Solving this will give \( \frac{a}{3}+\frac{b}{4}+\frac{c}{5}=\alpha^{2} \)

Solving the Obtained System of Equations

Now we have a system of equations with three equations and three variables. We can solve this system to find the values of \(a\), \(b\), and \(c\) that establish the function \(f(x)\).