https://file.aiquickdraw.com/mathbot/file/1727056196024y5j4sxj3.jpg

To ensure that the function \( f(x) \) is both continuous and differentiable, we need to ensure two conditions are met at the point \( x = -2 \):

1. **Continuity**: The function values from both parts must be equal at \( x = -2 \).
2. **Differentiability**: The derivatives from both parts must also be equal at \( x = -2 \).

### Step 1: Ensure Continuity
For continuity at \( x = -2 \):
\[ 4(-2)^3 - 7(-2)^2 + 3 = a(-2) + b \]

Simplifying each side:
\[ 4(-8) - 7(4) + 3 = -2a + b \]
\[ -32 - 28 + 3 = -2a + b \]
\[ -57 = -2a + b \]
This gives us our first equation.

### Step 2: Ensure Differentiability
To ensure differentiability, the derivatives must be equal at \( x = -2 \). The derivative of the cubic polynomial is:
\[ \frac{d}{dx}(4x^3 - 7x^2 + 3) = 12x^2 - 14x \]

Evaluating this at \( x = -2 \):
\[ 12(-2)^2 - 14(-2) = 12(4) + 28 = 76 \]

For \( ax + b \), the derivative is \( a \). So, at \( x = -2 \), the derivative from the linear part \( a \) must equal 76:
\[ a = 76 \]

### Step 3: Solve for \( b \)
Using \( a = 76 \) in our continuity equation:
\[ -57 = -2(76) + b \]
\[ -57 = -152 + b \]
\[ b = 95 \]

### Conclusion
The values of \( a \) and \( b \) that make \( f(x) \) both continuous and differentiable at \( x = -2 \) are \( a = 76 \) and \( b = 95 \).

Would you like more details on any part of the solution or have any further questions?

Here are some additional topics you might be interested in:
1. How to graph this piecewise function.
2. What happens to the function's properties if values other than \( a = 76 \) and \( b = 95 \) are used?
3. How can we find the limits of this function at other points?
4. Exploring the behavior of the function at extreme values of \( x \).
5. How continuity and differentiability influence the shape of a graph.

Tip: Always verify solutions by plugging the results back into the original conditions for both the function and its derivative.

Find a and b so that the function f(x) = {4x^3 - 7x^2 + 3, x < -2; ax + b, x >= -2} is both continuous and differentiable.

Solve the problem of finding the values of a and b that make a piecewise function continuous and differentiable at x = -2. This solution involves verifying continuity and matching derivatives to ensure differentiability.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation