Math Problem Statement
Given the function f(x) = {2x + 1, x ≤ 0 ; -x^2 + 2x + 2, x > 0}, which of the following assumptions is true?
Solution
Let's solve this step by step.
We are given the piecewise function:
\begin{cases} 2x + 1, & x \leq 0 \\ -x^2 + 2x + 2, & x > 0 \end{cases}$$ The question is about the differentiability of this function at $$x = 0$$. ### Step 1: Check Continuity at $$x = 0$$ For a function to be differentiable at a point, it must first be continuous there. So we need to check if $$f(x)$$ is continuous at $$x = 0$$. - For $$x \leq 0$$, $$f(x) = 2x + 1$$. $$\lim_{x \to 0^-} f(x) = 2(0) + 1 = 1$$ - For $$x > 0$$, $$f(x) = -x^2 + 2x + 2$$. $$\lim_{x \to 0^+} f(x) = -(0)^2 + 2(0) + 2 = 2$$ Since $$\lim_{x \to 0^-} f(x) = 1$$ and $$\lim_{x \to 0^+} f(x) = 2$$, the left-hand and right-hand limits are not equal. Hence, $$f(x)$$ is **not continuous** at $$x = 0$$. ### Step 2: Differentiability at $$x = 0$$ Because $$f(x)$$ is not continuous at $$x = 0$$, it cannot be differentiable at $$x = 0$$ either. Differentiability implies continuity, so if a function is not continuous at a point, it is automatically not differentiable there. ### Step 3: Conclusion The function is not differentiable at $$x = 0$$, meaning option **(c)** is correct: "**$$f$$ is not differentiable at $$x_0 = 0$$**." Let me know if you would like more details or have any further questions! --- Here are 5 questions to extend this understanding: 1. What is the definition of differentiability at a point? 2. Can a function be continuous but not differentiable at a point? 3. How do you determine the derivative of a piecewise function? 4. How would you check if the function is continuous for other values of $$x$$? 5. Why does differentiability imply continuity but not vice versa? **Tip:** Always check continuity before checking differentiability, as a function must be continuous at a point to be differentiable there.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Continuity
Differentiability
Piecewise Functions
Formulas
Limit Definition of Continuity
Definition of Differentiability
Theorems
Differentiability implies continuity
Suitable Grade Level
Grades 11-12
Related Recommendation
Differentiability and Continuity of Piecewise Functions at x = 1 and x = -1
Limit Analysis of a Piecewise Function with Potential Discontinuities
Piecewise Differentiability: Identify the Function Differentiable at x = 3
Continuity and Differentiability of a Piecewise Function
Determine the Continuity of a Piecewise Function at x = 0