Math Problem Statement
Discuss the differentiability of the function f(x) defined as: f(x) = ln(x) when 0 < x <= 1, 0 when 1 < x <= 2, and 1 + x^2 when x > 2, at x = 1.
Solution
The function is defined piecewise as follows:
\begin{cases} \ln x & \text{when } 0 < x \leq 1 \\ 0 & \text{when } 1 < x \leq 2 \\ 1 + x^2 & \text{when } x > 2 \end{cases}$$ We are asked to discuss the differentiability of the function at $$x = 1$$. ### Step 1: Check Continuity at $$x = 1$$ For a function to be differentiable at a point, it must first be continuous at that point. To check continuity at $$x = 1$$, we need to ensure the left-hand limit and the right-hand limit of $$f(x)$$ as $$x \to 1$$ are equal to the function value at $$x = 1$$. - Left-hand limit as $$x \to 1^-$$ (i.e., from the left): $$\lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} \ln x = \ln(1) = 0$$ - Right-hand limit as $$x \to 1^+$$ (i.e., from the right): $$\lim_{x \to 1^+} f(x) = 0$$ Since both the left-hand and right-hand limits are equal to 0, the function is continuous at $$x = 1$$. ### Step 2: Check Differentiability at $$x = 1$$ Next, we need to check if the derivative exists at $$x = 1$$. For this, we compute the left-hand and right-hand derivatives. - **Left-hand derivative as $$x \to 1^-$$**: $$f'(x) = \frac{d}{dx} (\ln x) = \frac{1}{x}$$ So, the left-hand derivative is: $$\lim_{x \to 1^-} f'(x) = \lim_{x \to 1^-} \frac{1}{x} = 1$$ - **Right-hand derivative as $$x \to 1^+$$**: The function for $$1 < x \leq 2$$ is constant (0), so the derivative is: $$f'(x) = 0$$ Since the left-hand derivative is 1 and the right-hand derivative is 0, the derivative does not match on both sides of $$x = 1$$. Therefore, $$f(x)$$ is **not differentiable** at $$x = 1$$. ### Conclusion: The function $$f(x)$$ is continuous but **not differentiable** at $$x = 1$$. --- Would you like more details or have any questions? Here are 5 related questions to explore: 1. What is the definition of continuity and how is it tested? 2. How do we compute derivatives for piecewise functions? 3. What happens if a function is continuous but not differentiable at a point? 4. Can you provide another example where a function is not differentiable at a specific point? 5. How does the concept of differentiability relate to the shape of a graph? **Tip**: Always check for continuity before testing differentiability at a point in a piecewise function.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Differentiability
Continuity
Piecewise Functions
Limits
Formulas
f'(x) = d/dx (ln x) = 1/x
Left-hand limit and right-hand limit
Left-hand and right-hand derivatives
Theorems
Differentiability implies continuity
Piecewise continuity and differentiability
Suitable Grade Level
College Level Calculus
Related Recommendation
Analyzing Continuity and Differentiability of a Piecewise Function at x = 1
One-Sided Derivatives and Differentiability at x = 1 for Piecewise Function
Study Continuity and Differentiability of a Piecewise Function f(x)
Differentiability of a Piecewise Function at x = 1
Differentiability Analysis of a Piecewise Function at x=0 and x=π/2