Math Problem Statement
Find the values of a and b that make the following function continuous at x = 1. f(x) = {2x + 3a if x < 1; a + 4 if x = 1; sqrt(bx) if x > 1}.
Solution
We are given the following piecewise function :
\begin{cases} 2x + 3a & \text{if } x < 1 \\ a + 4 & \text{if } x = 1 \\ \sqrt{bx} & \text{if } x > 1 \end{cases}$$ We need to find values of $$a$$ and $$b$$ such that the function is continuous at $$x = 1$$. ### Step 1: Condition for Continuity A function is continuous at $$x = 1$$ if: $$\lim_{x \to 1^-} f(x) = f(1) = \lim_{x \to 1^+} f(x)$$ ### Step 2: Left-hand limit $$\lim_{x \to 1^-} f(x)$$ For $$x < 1$$, $$f(x) = 2x + 3a$$. Thus, we compute: $$\lim_{x \to 1^-} f(x) = 2(1) + 3a = 2 + 3a$$ ### Step 3: Value at $$x = 1$$ From the function definition, $$f(1) = a + 4$$. ### Step 4: Right-hand limit $$\lim_{x \to 1^+} f(x)$$ For $$x > 1$$, $$f(x) = \sqrt{bx}$$. Thus, we compute: $$\lim_{x \to 1^+} f(x) = \sqrt{b(1)} = \sqrt{b}$$ ### Step 5: Set up the Continuity Conditions For the function to be continuous at $$x = 1$$, we need the following equalities: 1. $$\lim_{x \to 1^-} f(x) = f(1)$$, i.e., $$2 + 3a = a + 4$$ 2. $$f(1) = \lim_{x \to 1^+} f(x)$$, i.e., $$a + 4 = \sqrt{b}$$ ### Step 6: Solve the System of Equations 1. Solve $$2 + 3a = a + 4$$: $$2 + 3a = a + 4 \implies 2a = 2 \implies a = 1$$ 2. Substitute $$a = 1$$ into $$a + 4 = \sqrt{b}$$: $$1 + 4 = \sqrt{b} \implies \sqrt{b} = 5 \implies b = 25$$ ### Final Answer: The values of $$a$$ and $$b$$ that make the function continuous at $$x = 1$$ are: $$a = 1, \quad b = 25$$ Would you like further details or have any questions? Here are 5 related questions: 1. How can you prove that a function is continuous at a different point, such as $$x = 2$$? 2. What happens if the function was discontinuous at $$x = 1$$? What kind of discontinuity would it be? 3. How do you find the derivative of a piecewise function like this? 4. How does the square root function $$\sqrt{bx}$$ affect the overall continuity of the function? 5. How would the continuity condition change if we were dealing with limits approaching infinity? **Tip:** When checking for continuity at a point, always ensure that both the left-hand and right-hand limits match the function value at that point.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→1^-) f(x) = 2 + 3a
Value at x = 1: f(1) = a + 4
Right-hand limit: lim(x→1^+) f(x) = sqrt(b)
Theorems
Continuity theorem: A function is continuous at a point if the left-hand limit, the right-hand limit, and the function value at that point are equal.
Suitable Grade Level
College-level calculus
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