Math Problem Statement
Solution
Let's work through the problem step by step.
We are given the piecewise function:
\begin{cases} x^2 - 4x & \text{if } x \leq 6 \\ 2x & \text{if } x > 6 \end{cases}$$ ### (a) Is $$f$$ continuous at $$x = 6$$? To check continuity at $$x = 6$$, we need to verify if: $$\lim_{x \to 6^-} f(x) = \lim_{x \to 6^+} f(x) = f(6)$$ 1. **Left-hand limit** $$\lim_{x \to 6^-} f(x)$$: For $$x \leq 6$$, $$f(x) = x^2 - 4x$$. So, $$\lim_{x \to 6^-} f(x) = 6^2 - 4(6) = 36 - 24 = 12$$ 2. **Right-hand limit** $$\lim_{x \to 6^+} f(x)$$: For $$x > 6$$, $$f(x) = 2x$$. So, $$\lim_{x \to 6^+} f(x) = 2(6) = 12$$ 3. **Value at $$x = 6$$**: We use the piece $$f(x) = x^2 - 4x$$ since $$x = 6$$ is included in the first part of the piecewise function. So, $$f(6) = 6^2 - 4(6) = 12$$ Since the left-hand limit, right-hand limit, and $$f(6)$$ are all equal to 12, **$$f$$ is continuous at $$x = 6$$**. ### (b) Is $$f$$ differentiable at $$x = 6$$? To check differentiability, we need to verify if the left-hand derivative and right-hand derivative at $$x = 6$$ are equal. 1. **Left-hand derivative** (for $$x \leq 6$$): For $$f(x) = x^2 - 4x$$, the derivative is: $$f'(x) = 2x - 4$$ At $$x = 6$$, $$f'(6^-) = 2(6) - 4 = 12 - 4 = 8$$ 2. **Right-hand derivative** (for $$x > 6$$): For $$f(x) = 2x$$, the derivative is: $$f'(x) = 2$$ At $$x = 6$$, $$f'(6^+) = 2$$ Since the left-hand derivative $$f'(6^-) = 8$$ and the right-hand derivative $$f'(6^+) = 2$$ are not equal, **$$f$$ is not differentiable at $$x = 6$$**. ### (c) Since $$f$$ is not differentiable at $$x = 6$$, we enter the value of $$f(6)$$: From part (a), we found that: $$f(6) = 12$$ So the answer is $$f(6) = 12$$. --- Would you like further details or have any questions on this? Here are some related questions: 1. How do you define the conditions for continuity in a piecewise function? 2. What does it mean for a function to be differentiable at a point? 3. Can a function be continuous but not differentiable at a point? Why? 4. What are some common points of non-differentiability in piecewise functions? 5. How can we interpret the graphical behavior of a function that is continuous but not differentiable? **Tip:** When working with piecewise functions, always check continuity first before checking differentiability at the points where the function changes its rule.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Continuity
Differentiability
Formulas
f(x) = x^2 - 4x for x <= 6
f(x) = 2x for x > 6
Continuity condition: lim(x -> a-) f(x) = lim(x -> a+) f(x) = f(a)
Differentiability condition: f'(a-) = f'(a+)
Theorems
Continuity Theorem
Differentiability Theorem
Piecewise function analysis
Suitable Grade Level
Grades 11-12 (Pre-Calculus/Calculus)
Related Recommendation
Piecewise Function Continuity and Differentiability Problem
Analyzing Continuity and Differentiability of a Piecewise Function
Continuity and Non-Differentiability of a Piecewise Function at x = 1
Graph and Analyze Piecewise Function f(x) = 6, -10 + x^2, 2x - 2
Continuity and Differentiability of a Piecewise Function at x = 2