Math Problem Statement
Solution
The image describes a piecewise function defined as:
\begin{cases} x^2 - 4, & x \leq 2 \\ k(x - 2), & x > 2 \end{cases}$$ The questions are: ### (a) For what values of $$k$$ is $$f$$ continuous? For continuity at $$x = 2$$, the left-hand limit and the right-hand limit must be equal to the value of the function at $$x = 2$$. Let's evaluate this: 1. The left-hand limit as $$x \to 2^-$$ is: $$\lim_{x \to 2^-} f(x) = 2^2 - 4 = 0$$ 2. The right-hand limit as $$x \to 2^+$$ is: $$\lim_{x \to 2^+} f(x) = k(2 - 2) = 0$$ 3. The value of the function at $$x = 2$$ is $$f(2) = 2^2 - 4 = 0$$. Since all limits and the function value at $$x = 2$$ are $$0$$, the function is continuous for **all values of $$k$$**. ### (b) For what values of $$k$$ is $$f$$ differentiable? For differentiability at $$x = 2$$, the left-hand and right-hand derivatives must be equal. Let's compute both: 1. The derivative of $$f(x) = x^2 - 4$$ for $$x \leq 2$$ is: $$f'(x) = 2x$$ So the left-hand derivative as $$x \to 2^-$$ is: $$\lim_{x \to 2^-} f'(x) = 2(2) = 4$$ 2. The derivative of $$f(x) = k(x - 2)$$ for $$x > 2$$ is: $$f'(x) = k$$ So the right-hand derivative as $$x \to 2^+$$ is: $$\lim_{x \to 2^+} f'(x) = k$$ For $$f(x)$$ to be differentiable at $$x = 2$$, the left-hand and right-hand derivatives must be equal. Therefore: $$4 = k$$ Thus, $$f(x)$$ is differentiable when $$k = 4$$. --- Let me know if you'd like further explanation on any part of this solution! ### 5 Related Questions: 1. How do you check the continuity of a piecewise function at the boundary point? 2. What is the importance of left-hand and right-hand derivatives for differentiability? 3. How does the differentiability of a function imply continuity? 4. Can a function be continuous but not differentiable? If so, when? 5. What happens to the continuity or differentiability when $$k \neq 4$$? ### Tip: When solving for continuity and differentiability, always compute limits from both sides of the boundary point carefully, ensuring the function behavior is smooth (continuous) and the slope matches (differentiable).Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity
Differentiability
Piecewise Functions
Formulas
f'(x) = 2x for x ≤ 2
f'(x) = k for x > 2
Theorems
The function is continuous at x = 2 if the left-hand limit and right-hand limit are equal to the value at x = 2
The function is differentiable if the left-hand derivative equals the right-hand derivative at x = 2
Suitable Grade Level
Grades 11-12 or introductory college calculus
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