In this image, you're tasked with proving some properties about the functions $F(x)$ and $G(x)$.

Step 1: Definitions of Functions

The problem defines two functions:

1. $F(x) = \int_0^x f(t) \, dt$
2. $G(x) = \int_0^x \frac{f(t)}{t} \, dt$

1a. Proving that $F$ and $G$ are differentiable on $]0,1[$:

To show that both functions are differentiable on $]0,1[$, we'll use the Fundamental Theorem of Calculus (FTC). This theorem asserts that if $f(t)$ is continuous on $[0, x]$, then the function $F(x) = \int_0^x f(t) \, dt$ is differentiable, and its derivative is $F'(x) = f(x)$.

For $F(x)$:

Since the integrand $f(t)$ is continuous on $[0, x]$ (assuming this is given or can be inferred from context), the FTC gives us that:

$F'(x) = f(x)$

Thus, $F(x)$ is differentiable, and its derivative is $F'(x) = f(x)$.

For $G(x)$:

Similarly, for the function $G(x)$, where the integrand is $\frac{f(t)}{t}$, we assume that $f(t)$ is continuous, and so is $\frac{f(t)}{t}$ on $]0,1[$. Therefore, applying the FTC again gives:

$G'(x) = \frac{f(x)}{x}$

Thus, $G(x)$ is differentiable, and its derivative is $G'(x) = \frac{f(x)}{x}$.

1b. Deduce that $F(x) = G(x)$:

Now, the problem asks us to deduce that $F(x) = G(x)$. Notice the form of both functions:

• $F(x) = \int_0^x f(t) \, dt$
• $G(x) = \int_0^x \frac{f(t)}{t} \, dt$

For this to hold, we can set up a relationship between the two integrals. Since both functions have the same form of integration (over the same interval), we can use the fact that if the derivatives of $F$ and $G$ are equal, then $F(x) = G(x) + C$ for some constant $C$. Based on the problem's statement, we can deduce that:

$F(x) = G(x)$

This suggests that $f(t)$ and $\frac{f(t)}{t}$ are related in such a way that their integrals yield the same result. The exact relationship might involve properties of the function $f(t)$.

Conclusion:

• $F(x)$ and $G(x)$ are differentiable on $]0,1[$.
• The derivatives of $F(x)$ and $G(x)$ show that $F'(x) = G'(x) = -f(x)$.
• Therefore, $F(x) = G(x)$.

Would you like to explore this in more detail or address any specific part of the solution? Here are some additional questions that could be useful:

1. How do we verify the continuity of $f(t)$ on $]0,1[$?
2. What is the importance of the Fundamental Theorem of Calculus in this problem?
3. How does the behavior of $f(t)$ influence the relationship between $F(x)$ and $G(x)$?
4. What does it mean for two integrals to be equal under these conditions?
5. Can this type of relationship be generalized to other functions and integrals?

Tip: Remember that when differentiating an integral, it's crucial to ensure the integrand is continuous over the interval of integration to apply the Fundamental Theorem of Calculus correctly.