Let 𝑓f be the function

𝑓(𝑥)={𝑥2−𝑐4𝑥+2𝑐for 𝑥<5for 𝑥≥5
f(x)={x2−cfor x<54x+2cfor x≥5

Find the value of 𝑐c that makes the function continuous.

(Use symbolic notation and fractions where needed.)

To find the value of \( c \) that makes the function \( f(x) \) continuous, we need to ensure that the left-hand limit (as \( x \) approaches 5 from the left) equals the right-hand limit (as \( x \) approaches 5 from the right), and that both equal the function value at \( x = 5 \).

The function is defined as:

\[
f(x) = \begin{cases} 
x^2 - c & \text{for } x < 5 \\
4x + 2c & \text{for } x \geq 5 
\end{cases}
\]

### Step 1: Evaluate the limit as \( x \) approaches 5 from the left.
For \( x < 5 \), \( f(x) = x^2 - c \).

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

### Step 2: Evaluate the limit as \( x \) approaches 5 from the right.
For \( x \geq 5 \), \( f(x) = 4x + 2c \).

\[
\lim_{x \to 5^+} f(x) = 4(5) + 2c = 20 + 2c
\]

### Step 3: Set the limits equal to each other for continuity.
To make \( f(x) \) continuous at \( x = 5 \), the left-hand limit must equal the right-hand limit:

\[
25 - c = 20 + 2c
\]

### Step 4: Solve for \( c \).
\[
25 - 20 = 2c + c
\]
\[
5 = 3c
\]
\[
c = \frac{5}{3}
\]

Thus, the value of \( c \) that makes the function \( f(x) \) continuous is \( \boxed{\frac{5}{3}} \).

Would you like more details or have any questions? Here are some follow-up questions you might find interesting:

1. What does it mean for a function to be continuous at a point?
2. Can you describe the behavior of \( f(x) \) at \( x = 5 \) when \( c = \frac{5}{3} \)?
3. What happens if \( c \) takes a different value than \( \frac{5}{3} \)?
4. How would you find the value of \( c \) if the function was piecewise-defined over a different interval?
5. Can a function be continuous but not differentiable at a certain point?

**Tip:** Remember, for continuity at a point, the function must be defined at that point, and the left-hand and right-hand limits must both exist and be equal.

Explore how to find the value of c that ensures continuity for a piecewise function. This mathematical problem involves evaluating limits and setting equations to maintain function continuity at a critical point. Suitable for advanced high school students.

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