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

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
Transcript text: 17. [-/1 Points] DETAILS MY NOTES SCALC9 3.TF.017. Determine whether the statement is true or false. If $f$ is periodic and $f$ is differentiable, then $f^{\prime}$ is periodic. True False Need Help? Read It Submit Answer 18. [-/1 Points] DETAILS MY NOTES SCALC9 3.TF.018. Determine whether the statement is true or false. The most general antiderivative of $f(x)=3 x^{-2}$ is $F(x)=-\frac{3}{x}+C$. True False Need Help? Read It
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Solution

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Solution Steps

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

  2. For the second question, find the antiderivative of the function f(x)=3x2 f(x) = 3x^{-2} . The antiderivative is found by integrating the function and adding a constant of integration C C . Compare the result with the given expression to determine if it is correct.

Step 1: Analyze the Periodicity of the Derivative

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

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

Step 2: Find the Antiderivative

Next, we analyze the function f(x)=3x2 f(x) = 3x^{-2} to find its most general antiderivative. The integral of f(x) f(x) is calculated as follows:

F(x)=3x2dx=3x+C F(x) = \int 3x^{-2} \, dx = -\frac{3}{x} + C

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

Final Answer

The first statement is False, and the second statement is True.

Thus, the answers are:

  • For the first statement: \\(\boxed{\text{False}}\\)
  • For the second statement: \\(\boxed{\text{True}}\\)
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