Math Problem Statement
Consider the function f(x) = { c^2 * x^2, if x < 4; 16cx + 80, if x ≥ 4 } where c is a constant. Find all value(s) of the constant c such that f(x) is continuous at x = 4. List the values of c that you find separated by commas.
Solution
To solve the problem and find the value of such that the function is continuous at , we must ensure that the left-hand limit of as equals the right-hand limit as , and also that these limits equal the value of .
Step 1: Analyze the given function
The function is defined piecewise as:
\begin{cases} c^2 x^2 & \text{if } x < 4 \\ 16cx + 80 & \text{if } x \geq 4 \end{cases}$$ To ensure continuity at $$x = 4$$, the following condition must hold: $$\lim_{x \to 4^-} f(x) = \lim_{x \to 4^+} f(x) = f(4)$$ ### Step 2: Left-hand limit ($$x \to 4^-$$) For $$x < 4$$, we have $$f(x) = c^2 x^2$$. So, the left-hand limit as $$x \to 4^-$$ is: $$\lim_{x \to 4^-} f(x) = \lim_{x \to 4^-} c^2 x^2 = c^2 \cdot 4^2 = 16c^2$$ ### Step 3: Right-hand limit ($$x \to 4^+$$) For $$x \geq 4$$, we have $$f(x) = 16cx + 80$$. So, the right-hand limit as $$x \to 4^+$$ is: $$\lim_{x \to 4^+} f(x) = 16c \cdot 4 + 80 = 64c + 80$$ ### Step 4: Continuity condition For continuity at $$x = 4$$, we set the left-hand limit equal to the right-hand limit: $$16c^2 = 64c + 80$$ ### Step 5: Solve the equation We now solve the quadratic equation: $$16c^2 - 64c - 80 = 0$$ Divide through by 16 to simplify: $$c^2 - 4c - 5 = 0$$ Factor the quadratic equation: $$(c - 5)(c + 1) = 0$$ So, the solutions are: $$c = 5 \quad \text{or} \quad c = -1$$ Thus, the values of $$c$$ that make $$f(x)$$ continuous at $$x = 4$$ are $$c = 5$$ and $$c = -1$$. ### Final Answer: $$c = 5, -1$$ Would you like more details on any specific part of this solution? Here are some related questions to explore: 1. What happens to the function if $$c$$ takes any other value? 2. How does differentiability at $$x = 4$$ compare with continuity for this function? 3. Can you solve for continuity at other points (besides $$x = 4$$)? 4. How does the quadratic factorization method work? 5. What are the general conditions for piecewise functions to be continuous? **Tip**: When solving continuity problems, always set the left-hand and right-hand limits equal and make sure they match the function's value at the given point.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity
Piecewise Functions
Limits
Formulas
Left-hand limit = Right-hand limit = f(4)
Quadratic equation ax^2 + bx + c = 0
Theorems
Continuity of Piecewise Functions
Solving Quadratic Equations
Suitable Grade Level
Grades 10-12