Math Problem Statement
Solution
The problem asks us to determine whether the function is continuous and differentiable at . The function is defined as a piecewise function:
\begin{cases} x^2 & \text{if } x \leq 2 \\ 12 - x & \text{if } x > 2 \end{cases}$$ ### A. Check if $$f(x)$$ is continuous at $$x = 2$$: A function is continuous at $$x = 2$$ if: 1. $$f(2)$$ is defined. 2. The left-hand limit $$\lim_{x \to 2^-} f(x)$$ exists. 3. The right-hand limit $$\lim_{x \to 2^+} f(x)$$ exists. 4. Both limits are equal to $$f(2)$$. #### 1. Evaluate $$f(2)$$: Since $$2 \leq 2$$, use the first piece of the function: $$f(2) = 2^2 = 4.$$ #### 2. Evaluate the left-hand limit $$\lim_{x \to 2^-} f(x)$$: For $$x \leq 2$$, use $$f(x) = x^2$$: $$\lim_{x \to 2^-} f(x) = \lim_{x \to 2^-} x^2 = 2^2 = 4.$$ #### 3. Evaluate the right-hand limit $$\lim_{x \to 2^+} f(x)$$: For $$x > 2$$, use $$f(x) = 12 - x$$: $$\lim_{x \to 2^+} f(x) = \lim_{x \to 2^+} (12 - x) = 12 - 2 = 10.$$ #### 4. Compare limits and $$f(2)$$: $$\lim_{x \to 2^-} f(x) = 4, \quad \lim_{x \to 2^+} f(x) = 10, \quad f(2) = 4.$$ Since the left-hand limit is not equal to the right-hand limit, the function is **not continuous** at $$x = 2$$. ### B. Check if $$f(x)$$ is differentiable at $$x = 2$$: A function is differentiable at $$x = 2$$ only if it is continuous there. Since $$f(x)$$ is not continuous at $$x = 2$$, it is also **not differentiable** at $$x = 2$$. ### Conclusion: - **A.** $$f(x)$$ is **not continuous** at $$x = 2$$. - **B.** $$f(x)$$ is **not differentiable** at $$x = 2$$. Would you like a more detailed explanation or have any questions? Here are some related questions that might help you understand this topic better: 1. How do you determine if a function is continuous at a point? 2. What are the necessary conditions for a function to be differentiable at a point? 3. How do you find the left-hand and right-hand limits of a piecewise function? 4. What is the difference between continuity and differentiability? 5. Can a function be differentiable if it is not continuous? **Tip:** A function must be continuous at a point to be differentiable there, but continuity does not necessarily imply differentiability.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity
Differentiability
Piecewise Functions
Formulas
f(x) is continuous at x = 2 if lim(x -> 2-) f(x) = lim(x -> 2+) f(x) = f(2)
f(x) is differentiable at x = 2 if f(x) is continuous at x = 2 and the left-hand derivative equals the right-hand derivative
Theorems
Definition of Continuity
Definition of Differentiability
Suitable Grade Level
Grades 11-12
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