assume d/dx f(x)=f(x) which of the following statements are true?
1. F(x) + C is also an antiderivative of f(x) for any constant C.
2. F(x) is also an antiderivative of f(x) + C for any constant C
3. F(x+C) is also an antiderivative for f(x) for any constant C
4. F(x) is also an antiderivative for f(x+C) for any constant C
5. F(x) is an antiderivative of f(x).

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 \(C\).
5. \(F(x)\) 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.

Explore the implications of the differential equation d/dx f(x) = f(x) and its relationship to antiderivatives. Analyze statements regarding antiderivatives of f(x) under different conditions. Suitable for advanced high school students studying calculus.

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