Analyzing Piecewise Function f(x) with Limits and Derivatives

Math Problem Statement

Solution

Given the piecewise function $f(x)$ , let's analyze it:

x^3 + 5 & \text{if } x < 1, \\ 7x - x^3 & \text{if } x \geq 1. \end{cases}$$ ### 1. **Continuity at $$x = 1$$:** To check whether $$f(x)$$ is continuous at $$x = 1$$, we need to verify that the left-hand limit, the right-hand limit, and the value of the function at $$x = 1$$ are all equal. - **Left-hand limit** as $$x$$ approaches 1: $$\lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} (x^3 + 5) = 1^3 + 5 = 6$$ - **Right-hand limit** as $$x$$ approaches 1: $$\lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} (7x - x^3) = 7(1) - 1^3 = 7 - 1 = 6$$ - **Value of the function** at $$x = 1$$: $$f(1) = 7(1) - 1^3 = 7 - 1 = 6$$ Since the left-hand limit, the right-hand limit, and the value of the function at $$x = 1$$ are all equal, $$f(x)$$ is continuous at $$x = 1$$. ### 2. **Differentiability at $$x = 1$$:** Now let's check whether the derivative exists at $$x = 1$$. - **Left-hand derivative** as $$x$$ approaches 1: $$f'(x) = \frac{d}{dx}(x^3 + 5) = 3x^2$$ $$\lim_{x \to 1^-} f'(x) = 3(1)^2 = 3$$ - **Right-hand derivative** as $$x$$ approaches 1: $$f'(x) = \frac{d}{dx}(7x - x^3) = 7 - 3x^2$$ $$\lim_{x \to 1^+} f'(x) = 7 - 3(1)^2 = 7 - 3 = 4$$ Since the left-hand derivative $$3$$ and the right-hand derivative $$4$$ are not equal, the derivative does not exist at $$x = 1$$. ### Conclusion: - $$f(x)$$ is continuous at $$x = 1$$, but it is not differentiable at $$x = 1$$. Would you like more details or have any further questions? --- Here are 5 related questions: 1. How do you determine whether a piecewise function is differentiable at a point where the definition changes? 2. Can you give another example of a piecewise function that is continuous but not differentiable at a certain point? 3. How can we graphically interpret the discontinuity in the derivative at a point? 4. What is the geometric interpretation of a function's continuity at a point? 5. Can a function be differentiable at a point if it is not continuous at that point? **Tip:** When dealing with piecewise functions, always check for both continuity and differentiability at the points where the function definition changes.

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

Limit definition
Derivative definition

Theorems

Intermediate Value Theorem

Suitable Grade Level

Grades 11-12

Related Recommendation

Piecewise Function with Continuity and Differentiability Analysis

Graph of Piecewise Function f(x) with Detailed Analysis

Step-by-Step Analysis of Piecewise Function with Limits and Continuity

Determine Continuity of a Piecewise Function f(x) with Different Expressions for x ≤ 1 and x > 1

Differentiability Analysis of a Piecewise Function at x=0 and x=π/2