Math Problem Statement
Please solve this question.
Solution
To solve this problem, we first need to understand the function and its derivative.
Step 1: Understanding the function
The function is defined as:
\begin{cases} x & \text{if } x \geq 0, \\ -x & \text{if } x < 0. \end{cases}$$ ### Step 2: Derivative of $$f(x) = |x|$$ The derivative of $$f(x)$$ with respect to $$x$$ is piecewise defined as well: $$f'(x) = \begin{cases} 1 & \text{if } x > 0, \\ -1 & \text{if } x < 0, \\ \text{undefined} & \text{if } x = 0. \end{cases}$$ Note that at $$x = 0$$, the derivative does not exist because there is a sharp corner (cusp) at $$x = 0$$ in the graph of $$|x|$$. ### Step 3: Analyze the options Now, let's analyze each of the options provided: - **(a) $$f'(x) = 1$$ if $$x > 0$$**: This is true since the derivative of $$|x|$$ is $$1$$ for positive $$x$$. - **(b) $$f'(x) = -1$$ if $$x > 0$$**: This is false. As noted above, $$f'(x) = 1$$ when $$x > 0$$, not $$-1$$. - **(c) $$f'(x) = 0$$ if $$x > 0$$**: This is false. The correct derivative for $$x > 0$$ is $$1$$, not $$0$$. - **(d) $$f'(x) = 1$$ if $$x < 0$$**: This is false. For $$x < 0$$, the derivative is $$-1$$. ### Conclusion: The correct answer is **(a) $$f'(x) = 1$$ if $$x > 0$$**. Would you like further details or have any questions? Here are some additional questions related to this topic: 1. What is the derivative of $$|x|$$ at $$x = 0$$? 2. How do we find the derivative of a piecewise function? 3. What does the graph of $$|x|$$ look like? 4. What is a cusp in a function's graph, and how does it affect differentiability? 5. How can you determine the derivative of an absolute value function without calculus? 6. Can a function be continuous but not differentiable? Provide an example. 7. How does the concept of the derivative apply to real-world problems involving absolute values? 8. What are other examples of functions that are not differentiable at certain points? **Tip:** When dealing with absolute value functions, always consider the piecewise definition for analysis and derivative calculation.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Absolute Value Function
Derivative
Formulas
f(x) = |x|
f'(x) for |x|
Theorems
-
Suitable Grade Level
High School