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