For what value of the constant c is the function f continuous on (−∞, ∞)?
f(x) = 

 	cx2 + 6x	    	if x < 5
x3 − cx		if x ≥ 5
c =

To determine the value of \( c \) that makes the function \( f(x) \) continuous on \( (-\infty, \infty) \), we need to ensure that the function is continuous at \( x = 5 \) since this is the point where the definition of \( f(x) \) changes.

### Given:
- For \( x < 5 \): \( f(x) = cx^2 + 6x \)
- For \( x \geq 5 \): \( f(x) = x^3 - cx \)

### Continuity condition:
For \( f(x) \) to be continuous at \( x = 5 \), the left-hand limit as \( x \to 5^- \) must equal the right-hand limit as \( x \to 5^+ \), and both must equal \( f(5) \).

1. **Left-hand limit (as \( x \to 5^- \)):**

\[
\lim_{x \to 5^-} f(x) = \lim_{x \to 5^-} (cx^2 + 6x) = c(5^2) + 6(5) = 25c + 30
\]

2. **Right-hand limit (as \( x \to 5^+ \)):**

\[
\lim_{x \to 5^+} f(x) = \lim_{x \to 5^+} (x^3 - cx) = 5^3 - c(5) = 125 - 5c
\]

For continuity at \( x = 5 \), these two limits must be equal:

\[
25c + 30 = 125 - 5c
\]

### Solve for \( c \):

\[
25c + 5c = 125 - 30
\]
\[
30c = 95
\]
\[
c = \frac{95}{30} = \frac{19}{6}
\]

Thus, the value of \( c \) that makes \( f(x) \) continuous on \( (-\infty, \infty) \) is \( \boxed{\frac{19}{6}} \).

Would you like more details or have any questions?

Here are five related questions:
1. How do you check for differentiability at \( x = 5 \) for this function?
2. What happens if \( c \) takes a different value than \( \frac{19}{6} \)?
3. Can you graph the function to visualize continuity at \( x = 5 \)?
4. What if the function had different expressions for \( f(x) \) around \( x = 5 \)?
5. How do you compute the derivative of piecewise functions?

**Tip:** Always check both left-hand and right-hand limits when testing for continuity at a specific point in piecewise functions.

For what value of the constant c is the function f continuous on (-∞, ∞)?
f(x) =
  cx^2 + 6x if x < 5
  x^3 − cx if x ≥ 5
c = ?

Explore how to find the value of c that ensures the continuity of the piecewise function f(x) across its domain. This problem involves evaluating limits and solving equations to ensure the function is continuous at x = 5. Suitable for advanced high school or early college students.

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