Math Problem Statement
Define a piecewise function f(x) and determine the correct statements among the given options for continuity, limits, and differentiability.
Solution
To analyze the piecewise function defined as:
\begin{cases} 4x + x^2 & \text{for } x \leq 2 \\ x^3 + 4 & \text{for } x > 2 \end{cases}$$ we need to determine the correct statements among the given options by examining the continuity, limits, and differentiability of $$f(x)$$. ### 1. Continuity at $$x = 2$$ To determine if $$f(x)$$ is continuous at $$x = 2$$, we must check if: - $$f(2)$$ exists. - $$\lim_{x \to 2^-} f(x) = \lim_{x \to 2^+} f(x)$$. - $$f(2) = \lim_{x \to 2} f(x)$$. #### Calculating $$f(2)$$ Using $$f(x) = 4x + x^2$$ for $$x \leq 2$$: $$f(2) = 4(2) + (2)^2 = 8 + 4 = 12$$ #### Calculating $$\lim_{x \to 2^-} f(x)$$ $$\lim_{x \to 2^-} f(x) = \lim_{x \to 2} (4x + x^2) = 4(2) + (2)^2 = 12$$ #### Calculating $$\lim_{x \to 2^+} f(x)$$ $$\lim_{x \to 2^+} f(x) = \lim_{x \to 2} (x^3 + 4) = (2)^3 + 4 = 8 + 4 = 12$$ Since $$f(2) = \lim_{x \to 2^-} f(x) = \lim_{x \to 2^+} f(x) = 12$$, the function $$f(x)$$ is continuous at $$x = 2$$. ### 2. Limit as $$x \to 3$$ The limit as $$x \to 3$$ for $$f(x)$$ can be directly evaluated using the second piece of the function ($$f(x) = x^3 + 4$$ for $$x > 2$$): $$\lim_{x \to 3} f(x) = 3^3 + 4 = 27 + 4 = 31$$ So, the limit $$\lim_{x \to 3} f(x)$$ exists. ### 3. Differentiability at $$x = 2$$ To check if $$f(x)$$ is differentiable at $$x = 2$$, we need to ensure that the left-hand derivative and right-hand derivative at $$x = 2$$ are equal. #### Left-hand derivative at $$x = 2$$ For $$x \leq 2$$, $$f(x) = 4x + x^2$$. $$f'(x) = 4 + 2x$$ $$\lim_{x \to 2^-} f'(x) = 4 + 2(2) = 4 + 4 = 8$$ #### Right-hand derivative at $$x = 2$$ For $$x > 2$$, $$f(x) = x^3 + 4$$. $$f'(x) = 3x^2$$ $$\lim_{x \to 2^+} f'(x) = 3(2)^2 = 3 \times 4 = 12$$ Since the left-hand derivative $$(8)$$ and the right-hand derivative $$(12)$$ at $$x = 2$$ are not equal, $$f(x)$$ is **not differentiable** at $$x = 2$$. ### Conclusion Based on the above analysis: - $$f(x)$$ is continuous. - $$\lim_{x \to 3} f(x)$$ exists. - $$f(x)$$ is **not differentiable** at $$x = 2$$. The correct answers are: - **$$f(x)$$ is continuous** - **$$f(x)$$ is not differentiable** Would you like more details on any specific part, or have any questions? --- Here are five related questions to deepen understanding: 1. How can we check the differentiability of a piecewise function at a boundary point? 2. What is the significance of left-hand and right-hand derivatives in determining differentiability? 3. Why does the continuity of a function not guarantee its differentiability? 4. How can piecewise functions be made differentiable at the boundary points? 5. What are the practical applications of piecewise functions in real-world problems? **Tip:** For piecewise functions, always check the boundary points for continuity and differentiability, as they often lead to discontinuities or non-differentiable points.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Continuity
Limits
Differentiability
Formulas
Continuity: f(a) = lim_{x \to a^-} f(x) = lim_{x \to a^+} f(x)
Limit: lim_{x \to a} f(x)
Differentiability: Left-hand derivative = Right-hand derivative at point
Theorems
Continuity Theorem
Differentiability Theorem
Suitable Grade Level
Grades 11-12