Math Problem Statement
Solution
To solve this problem, we need to ensure the function is continuous at . Continuity at requires that the left-hand limit and right-hand limit of as approaches 2 are equal, and that they also equal the function's value at .
Given the piecewise function:
cx^2 + 3x & \text{if } x < 2 \\ x^3 - cx & \text{if } x \geq 2 \end{cases}$$ ### Step 1: Find the Left-Hand Limit as $$x \to 2^-$$ The left-hand limit as $$x$$ approaches 2 from the left is: $$\lim_{x \to 2^-} f(x) = \lim_{x \to 2^-} (cx^2 + 3x)$$ Substitute $$x = 2$$ into the equation: $$\lim_{x \to 2^-} f(x) = c(2)^2 + 3(2) = 4c + 6$$ ### Step 2: Find the Right-Hand Limit as $$x \to 2^+$$ The right-hand limit as $$x$$ approaches 2 from the right is: $$\lim_{x \to 2^+} f(x) = \lim_{x \to 2^+} (x^3 - cx)$$ Substitute $$x = 2$$ into the equation: $$\lim_{x \to 2^+} f(x) = (2)^3 - c(2) = 8 - 2c$$ ### Step 3: Set the Limits Equal for Continuity For the function to be continuous at $$x = 2$$, the left-hand limit and right-hand limit must be equal: $$4c + 6 = 8 - 2c$$ Now, solve for $$c$$: $$4c + 2c = 8 - 6$$ $$6c = 2$$ $$c = \frac{1}{3}$$ So, the value of the constant $$c$$ that makes the function continuous on $$(-\infty, \infty)$$ is $$c = \frac{1}{3}$$. If you have any further questions or need more details, feel free to ask! --- ### Related Questions: 1. How does the value of $$c$$ affect the shape of the graph for $$x < 2$$ and $$x \geq 2$$? 2. What happens to the function if $$c$$ is not equal to $$\frac{1}{3}$$? 3. How do you check for differentiability at $$x = 2$$? 4. Can the function be discontinuous at any other point besides $$x = 2$$? 5. How would you find the derivative of the function for both intervals? ### Tip: Always remember to check both the left-hand and right-hand limits when verifying continuity at a particular point in a piecewise function.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity
Piecewise Functions
Limits
Formulas
Left-hand limit: lim(x → 2-) f(x) = lim(x → 2-) (cx^2 + 3x)
Right-hand limit: lim(x → 2+) f(x) = lim(x → 2+) (x^3 - cx)
Theorems
Continuity at a point theorem
Suitable Grade Level
Grades 10-12
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