Questions: 17. Determine whether the statement is true or false. If f is periodic and f is differentiable, then f' is periodic. True False 18. Determine whether the statement is true or false. The most general antiderivative of f(x)=3 x^(-2) is F(x)=-3/x+C. True False

Solution Steps

1. For the first question, consider the properties of periodic functions and their derivatives. A function \( f \) is periodic if there exists a period \( T \) such that \( f(x + T) = f(x) \) for all \( x \). Analyze whether the derivative \( f' \) retains this periodicity.

2. For the second question, find the antiderivative of the function \( f(x) = 3x^{-2} \). The antiderivative is found by integrating the function and adding a constant of integration \( C \). Compare the result with the given expression to determine if it is correct.

To determine whether the statement "If \( f \) is periodic and \( f \) is differentiable, then \( f' \) is periodic" is true, we consider the definition of periodic functions. A function \( f \) is periodic if there exists a period \( T \) such that \( f(x + T) = f(x) \) for all \( x \).

For example, if we take \( f(x) = \sin(x) \), which is periodic with period \( 2\pi \), its derivative \( f'(x) = \cos(x) \) is also periodic with the same period. However, if we consider a non-periodic function like \( f(x) = x \), its derivative \( f'(x) = 1 \) is constant and thus periodic with any period. Therefore, the statement is not universally true.

Next, we analyze the function \( f(x) = 3x^{-2} \) to find its most general antiderivative. The integral of \( f(x) \) is calculated as follows:

\[ F(x) = \int 3x^{-2} \, dx = -\frac{3}{x} + C \]

where \( C \) is the constant of integration. The given expression for the antiderivative is \( F(x) = -\frac{3}{x} + C \), which matches our calculated result.

Final Answer