Math Problem Statement
For what value of the constant b is the function f(x) continuous on (-∞, ∞), where:
f(x) = {
bx + 2, if x < 2
x^3 - bx, if x ≥ 2
}
Solution
The problem asks for the value of the constant such that the function is continuous on . The function is defined as:
\begin{cases} bx + 2, & \text{if } x < 2 \\ x^3 - bx, & \text{if } x \geq 2 \end{cases}$$ ### **Step-by-step solution:** A function is continuous at a point if the left-hand limit ($$\lim_{x \to c^-} f(x)$$), the right-hand limit ($$\lim_{x \to c^+} f(x)$$), and the function's value ($$f(c)$$) are all equal at that point. Here, we focus on the point $$x = 2$$, as this is where the definition of $$f(x)$$ changes. 1. **Left-hand limit ($$\lim_{x \to 2^-} f(x)$$):** For $$x < 2$$, the function is $$f(x) = bx + 2$$. As $$x \to 2^-$$: $$\lim_{x \to 2^-} f(x) = b(2) + 2 = 2b + 2$$ 2. **Right-hand limit ($$\lim_{x \to 2^+} f(x)$$):** For $$x \geq 2$$, the function is $$f(x) = x^3 - bx$$. As $$x \to 2^+$$: $$\lim_{x \to 2^+} f(x) = 2^3 - b(2) = 8 - 2b$$ 3. **Continuity condition:** For $$f(x)$$ to be continuous at $$x = 2$$, the left-hand limit and right-hand limit must be equal: $$\lim_{x \to 2^-} f(x) = \lim_{x \to 2^+} f(x)$$ Substituting the results: $$2b + 2 = 8 - 2b$$ 4. **Solve for $$b$$:** $$2b + 2b = 8 - 2$$ $$4b = 6$$ $$b = \frac{3}{2}$$ ### **Final Answer:** The value of $$b$$ such that $$f(x)$$ is continuous on $$(-\infty, \infty)$$ is: $$\boxed{b = \frac{3}{2}}$$ --- Let me know if you'd like further clarifications or have any questions! --- ### Related Questions: 1. What is the definition of continuity for piecewise functions at a point? 2. How do we determine if a piecewise function is differentiable at a point? 3. If $$b = \frac{3}{2}$$, what is $$f(2)$$? 4. How can we graphically interpret the continuity of this function? 5. What changes if the condition was $$f(x)$$ being differentiable instead of continuous? ### Tip: When solving for continuity in piecewise functions, always check both the left-hand and right-hand limits at the points where the function's definition changes.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity of functions
Piecewise functions
Limits
Formulas
Left-hand limit = Right-hand limit for continuity at a point
Theorems
Definition of continuity at a point
Suitable Grade Level
Grades 11-12
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