Math Problem Statement
Let f be the function defined above. Which of the following statements is true?
A) f is not differentiable at x = 5 because f is not continuous at x = 5.
B) f is not differentiable at x = 5 because the graph of f has a sharp corner at x = 5.
C) f is not differentiable at x = 5 because the graph of f has a vertical tangent at x = 5.
D) f is not differentiable at x = 5 because f is not defined at x = 5.
Solution
The function is defined piecewise as:
x^2 - 20 & \text{for } x < 5 \\ -x^2 + 20 & \text{for } x \geq 5 \end{cases}$$ ### Step 1: Check Continuity at $$x = 5$$ For $$f(x)$$ to be differentiable at $$x = 5$$, it must first be continuous at $$x = 5$$. Let's check if the left-hand limit ($$x \to 5^-$$) and the right-hand limit ($$x \to 5^+$$) agree, and if they match the value of $$f(5)$$. - **Left-hand limit** as $$x \to 5^-$$: $$\lim_{x \to 5^-} f(x) = \lim_{x \to 5^-} (x^2 - 20) = 5^2 - 20 = 25 - 20 = 5$$ - **Right-hand limit** as $$x \to 5^+$$: $$\lim_{x \to 5^+} f(x) = \lim_{x \to 5^+} (-x^2 + 20) = -(5^2) + 20 = -25 + 20 = -5$$ Since the left-hand limit is $$5$$ and the right-hand limit is $$-5$$, the limits do not match. Therefore, $$f(x)$$ is not continuous at $$x = 5$$. ### Conclusion: Since the function is not continuous at $$x = 5$$, it cannot be differentiable there. Therefore, **option A** is correct: - $$f$$ is not differentiable at $$x = 5$$ because $$f$$ is not continuous at $$x = 5$$. --- Would you like a detailed explanation of any particular part? Let me know! Here are 5 questions related to this topic: 1. What conditions must a function satisfy to be differentiable at a point? 2. How can you determine the continuity of a piecewise function at a specific point? 3. Why is continuity a necessary condition for differentiability? 4. What is the significance of one-sided limits in determining continuity at a point? 5. How does the presence of a sharp corner or cusp affect differentiability? **Tip:** Always check the continuity of a function at a point before checking for differentiability!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Calculus
Piecewise Functions
Differentiability
Continuity
Formulas
Left-hand and right-hand limits
Definition of continuity
Differentiability conditions
Theorems
A function must be continuous at a point to be differentiable at that point
Suitable Grade Level
Grades 11-12
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