Math Problem Statement
Let a and b represent real numbers. Find the values of a and b such that f is continuous on the entire real number line, where f(x) is defined as: f(x) = { ax^2 + x - b, x ≤ 2; ax + b, 2 < x < 5; 2ax - 7, x ≥ 5 }.
Solution
We are given a piecewise function:
\begin{cases} ax^2 + x - b, & x \leq 2 \\ ax + b, & 2 < x < 5 \\ 2ax - 7, & x \geq 5 \end{cases}$$ We need to find the values of $$a$$ and $$b$$ such that $$f(x)$$ is continuous on the entire real number line. For continuity, the left-hand limit and right-hand limit at $$x = 2$$ and $$x = 5$$ must be equal. ### Step 1: Continuity at $$x = 2$$ We require that: $$\lim_{x \to 2^-} f(x) = \lim_{x \to 2^+} f(x)$$ The left-hand limit as $$x \to 2$$ is: $$\lim_{x \to 2^-} f(x) = a(2)^2 + 2 - b = 4a + 2 - b$$ The right-hand limit as $$x \to 2^+$$ is: $$\lim_{x \to 2^+} f(x) = 2a + b$$ For continuity at $$x = 2$$, we set these equal: $$4a + 2 - b = 2a + b$$ Simplifying this equation: $$4a + 2 - b = 2a + b \implies 2a + 2 = 2b \implies a + 1 = b$$ Thus, we have the relation: $$b = a + 1$$ ### Step 2: Continuity at $$x = 5$$ We require that: $$\lim_{x \to 5^-} f(x) = \lim_{x \to 5^+} f(x)$$ The left-hand limit as $$x \to 5$$ is: $$\lim_{x \to 5^-} f(x) = 5a + b$$ The right-hand limit as $$x \to 5^+$$ is: $$\lim_{x \to 5^+} f(x) = 2a(5) - 7 = 10a - 7$$ For continuity at $$x = 5$$, we set these equal: $$5a + b = 10a - 7$$ Substitute $$b = a + 1$$ from the previous step: $$5a + (a + 1) = 10a - 7$$ Simplifying this equation: $$6a + 1 = 10a - 7$$ $$6a - 10a = -7 - 1$$ $$-4a = -8$$ $$a = 2$$ ### Step 3: Solve for $$b$$ Now that we know $$a = 2$$, substitute into the equation $$b = a + 1$$: $$b = 2 + 1 = 3$$ ### Final Answer: The values of $$a$$ and $$b$$ that make the function continuous are: $$a = 2, \quad b = 3$$ Would you like further clarification or details on any part of the solution? Here are 5 related questions for practice: 1. What does it mean for a function to be continuous at a point? 2. How would the solution change if the function had a different form in one of the intervals? 3. Can a function be continuous but not differentiable at a point? Give an example. 4. How would you verify the differentiability of this function at $$x = 2$$ or $$x = 5$$? 5. If one of the pieces of the function was cubic instead of quadratic, how would that affect the process? **Tip:** When solving continuity problems with piecewise functions, always check the limits from both sides of the breakpoints.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity
Piecewise Functions
Limits
Formulas
Continuity condition: lim x->c- f(x) = lim x->c+ f(x)
Solving equations for continuity at points of piecewise functions
Theorems
Limit Theorem
Continuity Definition
Suitable Grade Level
Grades 11-12, early college
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