https://file.aiquickdraw.com/mathbot/file/1726682421434dsctfy26.jpg

The problem in the image describes two functions, \( f(x) = 4x^2 + 2x - 8 \) and \( g(x) = x^3 - x + 10 \), and asks to find the slope of the tangent line where their graphs are tangent to each other (i.e., the slopes at that point are the same). Here's how we can solve it step by step:

### 1. Tangency condition
For the graphs to be tangent to each other at some point \( P(a, b) \), the functions must satisfy two conditions at that point:
- The function values must be equal: \( f(a) = g(a) \).
- The slopes (derivatives) must be equal: \( f'(a) = g'(a) \).

### 2. Find the derivatives of the functions
- The derivative of \( f(x) = 4x^2 + 2x - 8 \) is:
  \[
  f'(x) = 8x + 2
  \]
- The derivative of \( g(x) = x^3 - x + 10 \) is:
  \[
  g'(x) = 3x^2 - 1
  \]

### 3. Solve the system of equations
To find the point \( a \) where the graphs are tangent, we set up the following system of equations:
- \( f(a) = g(a) \) implies:
  \[
  4a^2 + 2a - 8 = a^3 - a + 10
  \]
- \( f'(a) = g'(a) \) implies:
  \[
  8a + 2 = 3a^2 - 1
  \]

Let's solve these two equations to find the value of \( a \), and then we can substitute that value to find the slope.

I will proceed with solving the system.The solution to the system of equations is \( a = 3 \).

### 4. Find the slope at \( a = 3 \)
Now that we know \( a = 3 \), we can substitute this into the derivative of either function to find the slope of the tangent line at this point.

Using \( f'(x) = 8x + 2 \), let's calculate the slope at \( a = 3 \).The slope of the tangent line at the point where the graphs of \( f(x) \) and \( g(x) \) are tangent is 26. Thus, the correct answer is **B) 26**.

Do you have any questions or would you like further details?

### 5 Related Questions:
1. How do you determine if two functions are tangent to each other?
2. What are the steps to solve a system of non-linear equations involving functions and their derivatives?
3. Can two functions have more than one tangent point?
4. How do you interpret the derivative geometrically as the slope of a tangent line?
5. What happens if the second derivatives of the functions are equal at the tangent point?

**Tip:** The slope of a tangent line at a point on a curve is the instantaneous rate of change at that point, which is why we use the derivative to find it.

Find the slope of the tangent line where the graphs of the functions f(x) = 4x^2 + 2x - 8 and g(x) = x^3 - x + 10 touch each other (i.e., are tangent).

Learn how to find the slope of the tangent line at the point where the graphs of f(x) = 4x^2 + 2x - 8 and g(x) = x^3 - x + 10 are tangent. This problem involves solving a system of non-linear equations and applying differentiation techniques. Suitable for Grades 11-12.

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