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We are given a piecewise function \( f(x) \) defined as:

\[
f(x) = 
\begin{cases} 
cx^2 + 6x & \text{if } x < 5 \\
x^3 - cx & \text{if } x \geq 5 
\end{cases}
\]

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.

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}

Learn how to determine the value of the constant c that makes the function f(x) continuous on (-∞, ∞). This problem involves solving for continuity by equating the left-hand and right-hand limits of a piecewise function at a critical point, x = 5.

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