Math Problem Statement
Find the values of constant c that makes π(π) continuous: π(π₯)={π₯2βπ2 if π₯<4ππ₯+20 if π₯β₯4.
Solution
To determine the values of the constant that make the function
\begin{cases} x^2 - c^2 & \text{if } x < 4 \\ 4cx + 20 & \text{if } x \geq 4 \end{cases}$$ continuous, we need to ensure that the left-hand limit as $$x$$ approaches 4 is equal to the right-hand limit at $$x = 4$$, and both of these are equal to $$f(4)$$. 1. **Calculate the left-hand limit as $$x$$ approaches 4:** $$\lim_{x \to 4^-} f(x) = 4^2 - c^2 = 16 - c^2$$ 2. **Calculate the right-hand limit as $$x$$ approaches 4:** $$\lim_{x \to 4^+} f(x) = 4c(4) + 20 = 16c + 20$$ 3. **Set the limits equal to each other for continuity:** $$16 - c^2 = 16c + 20$$ 4. **Rearrange the equation:** $$-c^2 - 16c + 16 - 20 = 0 \\ -c^2 - 16c - 4 = 0$$ Multiplying the entire equation by -1 gives: $$c^2 + 16c + 4 = 0$$ 5. **Use the quadratic formula to solve for $$c$$:** $$c = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-16 \pm \sqrt{16^2 - 4 \cdot 1 \cdot 4}}{2 \cdot 1} = \frac{-16 \pm \sqrt{256 - 16}}{2} = \frac{-16 \pm \sqrt{240}}{2}$$ Simplifying further: $$c = \frac{-16 \pm 4\sqrt{15}}{2} = -8 \pm 2\sqrt{15}$$ Thus, the values of $$c$$ that make $$f(x)$$ continuous are: $$c = -8 + 2\sqrt{15} \quad \text{and} \quad c = -8 - 2\sqrt{15}$$ ### Summary of Results: - $$c = -8 + 2\sqrt{15}$$ - $$c = -8 - 2\sqrt{15}$$ Would you like any further details or have any questions? Here are some related questions you might consider: 1. What is the value of $$c$$ approximately? 2. How does changing the value of $$c$$ affect the function's graph? 3. Can you show how to plot the function $$f(x)$$ for different values of $$c$$? 4. What happens to the continuity of $$f(x)$$ if $$c$$ is not one of these values? 5. Can you explain how the quadratic formula works in this context? **Tip:** Always verify your solutions by substituting them back into the original conditions to check for continuity.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Calculus
Limits
Piecewise Functions
Continuity
Formulas
Limit as x approaches a
Quadratic formula
Theorems
Theorem on continuity of piecewise functions
Suitable Grade Level
Grades 11-12