Math Problem Statement
Find the values of a and b that make f continuous on (-∞, ∞), where f(x) is a piecewise function:
1. f(x) = (x^3 - 8) / (x - 2) for x < 2
2. f(x) = ax^3 - bx + 3 for 2 ≤ x < 3
3. f(x) = 2x - a + b for x ≥ 3
Solution
Let's solve the problem step by step to find the values of and that make the function continuous on .
Problem Breakdown:
The piecewise function is:
\begin{cases} \frac{x^3 - 8}{x - 2} & \text{if} \ x < 2 \\ ax^3 - bx + 3 & \text{if} \ 2 \leq x < 3 \\ 2x - a + b & \text{if} \ x \geq 3 \end{cases}$$ ### Step 1: Continuity at $$x = 2$$ For $$f(x)$$ to be continuous at $$x = 2$$: $$\lim_{x \to 2^-} f(x) = \lim_{x \to 2^+} f(x)$$ 1. Calculate $$\lim_{x \to 2^-} f(x)$$: $$\lim_{x \to 2^-} \frac{x^3 - 8}{x - 2} = \lim_{x \to 2^-} \frac{(x-2)(x^2 + 2x + 4)}{x - 2} = 2^2 + 2(2) + 4 = 12$$ 2. Calculate $$\lim_{x \to 2^+} f(x)$$: $$\lim_{x \to 2^+} (a(2)^3 - b(2) + 3) = 8a - 2b + 3$$ Equating the two: $$8a - 2b + 3 = 12$$ $$8a - 2b = 9$$ ### Step 2: Continuity at $$x = 3$$ For $$f(x)$$ to be continuous at $$x = 3$$: $$\lim_{x \to 3^-} f(x) = \lim_{x \to 3^+} f(x)$$ 1. Calculate $$\lim_{x \to 3^-} f(x)$$: $$a(3)^3 - b(3) + 3 = 27a - 3b + 3$$ 2. Calculate $$\lim_{x \to 3^+} f(x)$$: $$2(3) - a + b = 6 - a + b$$ Equating the two: $$27a - 3b + 3 = 6 - a + b$$ $$27a + a - 3b - b = 3$$ $$28a - 4b = 3$$ ### Step 3: Solve the System of Equations $$8a - 2b = 9$$ $$28a - 4b = 3$$ Solve by elimination or substitution: 1. Multiply the first equation by 2: $$16a - 4b = 18$$ 2. Subtract the second equation from the result: $$(16a - 4b) - (28a - 4b) = 18 - 3$$ $$-12a = 15$$ $$a = -\frac{15}{12} = -\frac{5}{4}$$ 3. Substitute $$a = -\frac{5}{4}$$ into the first equation: $$8(-\frac{5}{4}) - 2b = 9$$ $$-10 - 2b = 9$$ $$-2b = 19$$ $$b = -\frac{19}{2}$$ ### Final Answer: $$a = -\frac{5}{4}, \quad b = -\frac{19}{2}$$ Would you like a more detailed breakdown of any part of the solution? Here are some related questions: 1. How do you check if a piecewise function is differentiable? 2. How do you find the left and right-hand limits for complex functions? 3. What are the general conditions for a function to be continuous at a point? 4. How do you handle discontinuities of rational functions? 5. What is the significance of matching derivatives at boundary points? **Tip:** When dealing with piecewise functions, always remember to check both continuity and differentiability at the boundaries for smooth transitions.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Limits
Continuity of Functions
Polynomial Algebra
Formulas
Limit as x approaches a value
Equating left-hand and right-hand limits for continuity
Theorems
Continuity Definition: lim x→c- f(x) = lim x→c+ f(x) = f(c)
Suitable Grade Level
Grades 11-12 or early college (calculus level)
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