Determine the values of a and b which would result in the function f(x) being differentiable at
X = 1.
f() =
Зах + 1
3622
- 3x - 9
for for
X < 1
X > 1

To ensure the function \( f(x) \) is differentiable at \( x = 1 \), we must first ensure that it is continuous at \( x = 1 \) and that the derivative from both sides at \( x = 1 \) is equal.

Given:

\[
f(x) =
\begin{cases} 
3ax + 1 & \text{for } x < 1, \\
3bx^2 - 3x - 9 & \text{for } x > 1.
\end{cases}
\]

### Step 1: Continuity at \( x = 1 \)
For \( f(x) \) to be continuous at \( x = 1 \):

\[
\lim_{{x \to 1^-}} f(x) = \lim_{{x \to 1^+}} f(x) = f(1)
\]

**Left-hand limit:**

\[
\lim_{{x \to 1^-}} f(x) = 3a(1) + 1 = 3a + 1
\]

**Right-hand limit:**

\[
\lim_{{x \to 1^+}} f(x) = 3b(1)^2 - 3(1) - 9 = 3b - 3 - 9 = 3b - 12
\]

Setting these equal for continuity:

\[
3a + 1 = 3b - 12
\]

\[
3a - 3b = -13 \quad \text{(Equation 1)}
\]

### Step 2: Differentiability at \( x = 1 \)
For \( f(x) \) to be differentiable at \( x = 1 \), the derivatives from both sides must also be equal:

**Left-hand derivative:**

\[
f'(x) \text{ for } x < 1: \quad \frac{d}{dx} (3ax + 1) = 3a
\]

**Right-hand derivative:**

\[
f'(x) \text{ for } x > 1: \quad \frac{d}{dx} (3bx^2 - 3x - 9) = 6bx - 3
\]

At \( x = 1 \):

\[
f'(1) = 6b(1) - 3 = 6b - 3
\]

Setting these equal for differentiability:

\[
3a = 6b - 3
\]

\[
3a - 6b = -3 \quad \text{(Equation 2)}
\]

### Step 3: Solving the System of Equations
We now solve the system of equations:

1. \( 3a - 3b = -13 \)
2. \( 3a - 6b = -3 \)

Subtract Equation 1 from Equation 2:

\[
(3a - 6b) - (3a - 3b) = -3 - (-13)
\]

\[
-3b = 10
\]

\[
b = -\frac{10}{3}
\]

Substitute \( b = -\frac{10}{3} \) into Equation 1:

\[
3a - 3\left(-\frac{10}{3}\right) = -13
\]

\[
3a + 10 = -13
\]

\[
3a = -23
\]

\[
a = -\frac{23}{3}
\]

### Final Answer
The values of \( a \) and \( b \) that make the function \( f(x) \) differentiable at \( x = 1 \) are:

\[
a = -\frac{23}{3}, \quad b = -\frac{10}{3}
\]

---

Do you have any questions or would you like more details on any step?

### Related Questions:
1. How do you check if a function is continuous at a point?
2. What is the geometric interpretation of differentiability at a point?
3. How can the concept of limits help in understanding the continuity of functions?
4. What is the significance of having equal derivatives from both sides at a point?
5. Can you give an example of a function that is continuous but not differentiable at a point?

### Tip:
When solving systems of equations, it's helpful to simplify one equation at a time to make substitution or elimination straightforward.

Explore how to determine values of a and b to ensure differentiability of the function f(x) at x = 1. This detailed solution covers continuity requirements, derivative calculations, and solving a system of equations. Ideal for undergraduate students studying calculus and differential equations.

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