Find the value of ccc for which the function f(x)= { 2𝑥 2 + 3𝑐𝑥 + 5, 𝑥 < 1 𝑐𝑥 2 + 3𝑥 − 2, 𝑥 ≥ 1 is continuous.

To find the value of \( c \) that makes the function \( f(x) \) continuous, we need to ensure that the function does not have any jump discontinuities at \( x = 1 \). Specifically, the left-hand limit and right-hand limit of \( f(x) \) at \( x = 1 \) should be equal to the function's value at \( x = 1 \).

Given:
\[
f(x) = \begin{cases} 
2x^2 + 3cx + 5 & \text{for } x < 1, \\
cx^2 + 3x - 2 & \text{for } x \geq 1.
\end{cases}
\]

### Step 1: Calculate the left-hand limit as \( x \) approaches 1
The left-hand limit of \( f(x) \) as \( x \) approaches 1 is:
\[
\lim_{x \to 1^-} f(x) = 2(1)^2 + 3c(1) + 5 = 2 + 3c + 5 = 3c + 7.
\]

### Step 2: Calculate the right-hand limit as \( x \) approaches 1
The right-hand limit of \( f(x) \) as \( x \) approaches 1 is:
\[
\lim_{x \to 1^+} f(x) = c(1)^2 + 3(1) - 2 = c + 3 - 2 = c + 1.
\]

### Step 3: Set the left-hand limit equal to the right-hand limit
For \( f(x) \) to be continuous at \( x = 1 \), we set the left-hand limit equal to the right-hand limit:
\[
3c + 7 = c + 1.
\]

### Step 4: Solve for \( c \)
To solve for \( c \), subtract \( c \) from both sides:
\[
2c + 7 = 1.
\]
Now, subtract 7 from both sides:
\[
2c = -6.
\]
Finally, divide by 2:
\[
c = -3.
\]

### Conclusion:
The value of \( c \) that makes the function \( f(x) \) continuous at \( x = 1 \) is \( c = -3 \).

Would you like further details or have any questions?

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Here are 5 related questions:

1. How do you determine if a function is continuous at a specific point?
2. What is the significance of matching left-hand and right-hand limits in continuity problems?
3. How would the solution change if the piecewise function had a different form?
4. Can you have a situation where \( c \) could result in discontinuity? What would that look like?
5. How can you check continuity for more complex piecewise functions involving trigonometric functions?

**Tip:** Always remember to check the function's value at the point where you're testing for continuity, as it must match both the left-hand and right-hand limits.

Discover how to find the value of c that ensures continuity for a piecewise function f(x). This math problem involves calculating left-hand and right-hand limits and setting them equal to determine the suitable value of c. Suitable for advanced high school students in Grades 11-12.

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