Math Problem Statement
For what value of the constant c is the function f continuous on (-∞, ∞)? The function f(x) is defined as:
f(x) =
{cx^2 + 6x if x < 5
x^3 - cx if x ≥ 5}
Solution
We are given a piecewise function defined as:
\begin{cases} cx^2 + 6x & \text{if } x < 5 \\ x^3 - cx & \text{if } x \geq 5 \end{cases}$$ We are tasked with finding the value of the constant $$c$$ that makes the function continuous over $$(-\infty, \infty)$$. ### Conditions for Continuity For the function to be continuous on all of $$(-\infty, \infty)$$, the left-hand limit and the right-hand limit of $$f(x)$$ must equal the value of $$f(x)$$ at $$x = 5$$. This means: $$\lim_{x \to 5^-} f(x) = \lim_{x \to 5^+} f(x) = f(5)$$ ### Step 1: Left-hand limit $$\lim_{x \to 5^-} f(x)$$ For $$x < 5$$, we use $$f(x) = cx^2 + 6x$$. Substitute $$x = 5$$ into this expression: $$f(5^-) = c(5^2) + 6(5) = 25c + 30$$ ### Step 2: Right-hand limit $$\lim_{x \to 5^+} f(x)$$ For $$x \geq 5$$, we use $$f(x) = x^3 - cx$$. Substitute $$x = 5$$ into this expression: $$f(5^+) = 5^3 - c(5) = 125 - 5c$$ ### Step 3: Set the limits equal for continuity To ensure continuity at $$x = 5$$, set the left-hand limit equal to the right-hand limit: $$25c + 30 = 125 - 5c$$ ### Step 4: Solve for $$c$$ $$25c + 5c = 125 - 30$$ $$30c = 95$$ $$c = \frac{95}{30} = \frac{19}{6}$$ Thus, the value of $$c$$ that makes the function continuous is $$c = \frac{19}{6}$$. Would you like further details on this solution? Here are 5 related questions for deeper understanding: 1. What is the definition of a continuous function? 2. How do we check continuity at a point for a piecewise function? 3. Can a piecewise function be continuous at more than one point of transition? 4. What happens if we get different left-hand and right-hand limits in a continuity problem? 5. How does differentiability relate to continuity in such cases? **Tip:** Always double-check that the function values and limits match at the transition point for continuity.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity of Piecewise Functions
Algebra
Limits
Formulas
Continuity condition: lim x→a⁻ f(x) = lim x→a⁺ f(x)
Piecewise function evaluation for limits
Theorems
Continuity Theorem
Suitable Grade Level
Grades 11-12 or Early University Level