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It should be differentiable Witherspoon x=2. So what is a and b to make this possible?

To ensure the function \( f(x) \) is differentiable at \( x = 2 \), we need to make sure the function is both continuous and has the same derivative from both sides of \( x = 2 \).

The piecewise function is given as:

\[
f(x) = 
\begin{cases} 
x^2 - 2x - 7 & \text{for } x < 2 \\
ax^2 + b & \text{for } x \geq 2
\end{cases}
\]

### 1. Continuity at \( x = 2 \)

For the function to be continuous at \( x = 2 \), the left-hand limit as \( x \) approaches 2 must equal the right-hand limit, and they both must equal \( f(2) \).

- For \( x < 2 \), \( f(x) = x^2 - 2x - 7 \).
  - Evaluate \( f(2) \):
    \[
    f(2) = 2^2 - 2(2) - 7 = 4 - 4 - 7 = -7
    \]

- For \( x \geq 2 \), \( f(x) = ax^2 + b \).
  - At \( x = 2 \), \( f(2) = a(2^2) + b = 4a + b \).

Set the two equal for continuity:
\[
4a + b = -7 \quad \text{(Equation 1)}
\]

### 2. Differentiability at \( x = 2 \)

For differentiability, the left-hand derivative at \( x = 2 \) must equal the right-hand derivative at \( x = 2 \).

- The derivative of \( f(x) = x^2 - 2x - 7 \) (for \( x < 2 \)) is:
  \[
  f'(x) = 2x - 2
  \]
  At \( x = 2 \), the left-hand derivative is:
  \[
  f'(2) = 2(2) - 2 = 4 - 2 = 2
  \]

- The derivative of \( f(x) = ax^2 + b \) (for \( x \geq 2 \)) is:
  \[
  f'(x) = 2ax
  \]
  At \( x = 2 \), the right-hand derivative is:
  \[
  f'(2) = 2a(2) = 4a
  \]

Set the derivatives equal for differentiability:
\[
4a = 2 \quad \text{(Equation 2)}
\]

### 3. Solve the System of Equations

From Equation 2:
\[
4a = 2 \quad \Rightarrow \quad a = \frac{2}{4} = \frac{1}{2}
\]

Substitute \( a = \frac{1}{2} \) into Equation 1:
\[
4\left(\frac{1}{2}\right) + b = -7 \quad \Rightarrow \quad 2 + b = -7 \quad \Rightarrow \quad b = -9
\]

### Conclusion:

For the function to be differentiable at \( x = 2 \), the values of \( a \) and \( b \) must be:

\[
a = \frac{1}{2}, \quad b = -9
\]

Would you like more details on how differentiability is ensured, or any questions on this process?

### Related Questions:

1. How do we check if a function is continuous at a point?
2. What are the conditions for a function to be differentiable at a point?
3. How do you calculate the left-hand and right-hand limits of a function?
4. Can a function be continuous but not differentiable at a point? If yes, provide examples.
5. How would the solution change if the function were defined differently for \( x > 2 \)?

### Tip:
When solving for differentiability, always ensure the function is continuous at the point in question first, as differentiability implies continuity but not the other way around!

It should be differentiable at x=2. So what is a and b to make this possible?

Learn how to solve a piecewise function problem to ensure differentiability at x = 2. The solution covers concepts like continuity and differentiability, with specific focus on matching the function's values and derivatives from both sides of x = 2. Ideal for students at calculus level.

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