For what value of the constant \( c \) is the function \( f \) continuous on \( (-\infty, \infty) \) ? \[ f(x)=\left\{\begin{array}{ll}c x^{2}+2 x & \text { if } x<3 \\ x^{3}-c x & \text { if } x \geq 3\end{array}\right. \] Need Help? Watch it SUBMIT ANSWER

To ensure that the function \( f(x) \) is continuous at \( x = 3 \), we need the two pieces of the function to equal each other at that point. Thus, we must find \( c \) such that: \[ c(3^2) + 2(3) = 3^3 - c(3) \] This simplifies to: \[ 9c + 6 = 27 - 3c \] Now, combine like terms: \[ 9c + 3c = 27 - 6 \] \[ 12c = 21 \implies c = \frac{21}{12} = \frac{7}{4} \] So, the value of \( c \) for which the function is continuous on \( (-\infty, \infty) \) is \( \frac{7}{4} \). --- Did you know that piecewise functions often model real-life situations, like a person's speed changing when they hit a speed bump? It leads to a sudden change in motion, much like how these functions "switch gears" depending on the input value! Continuity is a fundamental concept in calculus and analysis, playing a pivotal role in understanding limits and integration. If you want to dive deeper, consider exploring textbooks like "Calculus" by James Stewart or "Principles of Mathematical Analysis" by Walter Rudin to unravel the fascinating world of continuous functions!