Let's analyze each statement given the condition $\frac{d}{dx} f(x) = f(x)$.

First, let's recall that if $F(x)$ is an antiderivative of $f(x)$, then $F'(x) = f(x)$.

Statement Analysis:

1. $F(x) + C$ is also an antiderivative of $f(x)$ for any constant $C$:

   • If $F(x)$ is an antiderivative of $f(x)$, then $F'(x) = f(x)$.
   • The derivative of $F(x) + C$ is $F'(x) + 0 = f(x)$.
   • Therefore, $F(x) + C$ is indeed an antiderivative of $f(x)$.
   • This statement is true.

2. $F(x)$ is also an antiderivative of $f(x) + C$ for any constant $C$:

   • If $F(x)$ is an antiderivative of $f(x)$, then $F'(x) = f(x)$.
   • For $F(x)$ to be an antiderivative of $f(x) + C$, it must be true that $F'(x) = f(x) + C$.
   • Since $F'(x) = f(x)$ and not $f(x) + C$, this statement is not true.
   • This statement is false.

3. $F(x+C)$ is also an antiderivative for $f(x)$ for any constant $C$:

   • Let $G(x) = F(x + C)$.
   • The derivative of $G(x)$ with respect to $x$ is $G'(x) = F'(x + C) = f(x + C)$.
   • For $G(x)$ to be an antiderivative of $f(x)$, we need $G'(x) = f(x)$, but we have $G'(x) = f(x + C)$.
   • Since $f(x + C) \neq f(x)$, this statement is not true.
   • This statement is false.

4. $F(x)$ is also an antiderivative for $f(x+C)$ for any constant $C$:

   • If $F(x)$ is an antiderivative of $f(x)$, then $F'(x) = f(x)$.
   • For $F(x)$ to be an antiderivative of $f(x + C)$, it must be true that $F'(x) = f(x + C)$.
   • Since $F'(x) = f(x)$ and not $f(x + C)$, this statement is not true.
   • This statement is false.

5. $F(x)$ is an antiderivative of $f(x)$:

   • By definition, if $F(x)$ is an antiderivative of $f(x)$, then $F'(x) = f(x)$.
   • This is the given condition $\frac{d}{dx} f(x) = f(x)$.
   • This statement is true.

Summary:

The true statements are:

1. $F(x) + C$ is also an antiderivative of $f(x)$ for any constant $C$.
2. $F(x)$ is an antiderivative of $f(x)$.

Additional Questions:

1. How do you find the antiderivative of a function?
2. What is the significance of the constant $C$ in the context of antiderivatives?
3. Can a function have more than one antiderivative?
4. What is the general form of the solution to the differential equation $\frac{d}{dx}f(x) = f(x)$?
5. How does the chain rule apply to the derivatives of composite functions?
6. What is the relationship between indefinite integrals and antiderivatives?
7. How do you verify if a function is an antiderivative of another function?
8. What is the importance of initial conditions in determining a specific antiderivative?

Tip:

When dealing with differential equations, always consider the role of arbitrary constants in the general solution, as they represent an entire family of solutions rather than a unique one.