Find k such that the line is tangent to the graph of the function.
Function    	Line
f(x) = kx3	y = 7x + 6

We are tasked with finding the value of \( k \) such that the line \( y = 7x + 6 \) is tangent to the graph of the function \( f(x) = kx^3 \).

### Steps:

1. **The condition for tangency**:  
   For a line to be tangent to a curve at some point \( x = a \), two conditions must hold:
   - The function values must be equal: \( f(a) = 7a + 6 \).
   - The slopes (derivatives) must be equal: \( f'(a) = 7 \).

2. **Find the derivative of \( f(x) \)**:  
   The derivative of \( f(x) = kx^3 \) is:
   \[
   f'(x) = 3kx^2
   \]
   At \( x = a \), the slope of the function is:
   \[
   f'(a) = 3ka^2
   \]

3. **Set the slopes equal**:  
   Since the slope of the tangent line is 7, we have:
   \[
   3ka^2 = 7
   \]
   Solving for \( k \), we get:
   \[
   k = \frac{7}{3a^2}
   \]

4. **Set the function values equal**:  
   The function values at \( x = a \) must also be equal, so:
   \[
   f(a) = k a^3 = 7a + 6
   \]
   Substituting \( k = \frac{7}{3a^2} \) into the equation:
   \[
   \frac{7}{3a^2} \cdot a^3 = 7a + 6
   \]
   Simplifying:
   \[
   \frac{7a}{3} = 7a + 6
   \]
   Multiply through by 3 to eliminate the fraction:
   \[
   7a = 21a + 18
   \]
   Rearranging the terms:
   \[
   -14a = 18
   \]
   Solving for \( a \):
   \[
   a = -\frac{9}{7}
   \]

5. **Substitute \( a \) into the equation for \( k \)**:  
   Now that we have \( a = -\frac{9}{7} \), substitute it into \( k = \frac{7}{3a^2} \):
   \[
   k = \frac{7}{3 \left( -\frac{9}{7} \right)^2} = \frac{7}{3 \cdot \frac{81}{49}} = \frac{7 \times 49}{243} = \frac{343}{243} = \frac{49}{35} = \frac{7}{5}
   \]

Thus, the value of \( k \) is \( \frac{7}{5} \).

Would you like more details on any part of this explanation?

Here are five follow-up questions:
1. How do you find the derivative of a cubic function like \( f(x) = kx^3 \)?
2. What is the geometrical interpretation of a line being tangent to a curve?
3. How would the problem change if the function were a higher degree polynomial?
4. What if the line had a different slope, say \( y = 8x + 6 \)? How would the solution change?
5. How can we check if our solution is correct by plotting?

**Tip:** Always double-check your algebraic manipulations, especially when dealing with fractions and powers.

Find k such that the line y = 7x + 6 is tangent to the graph of the function f(x) = kx^3.

Explore how to find the value of k such that the line y = 7x + 6 is tangent to the cubic function f(x) = kx^3. This problem involves key concepts like derivatives, tangency conditions, and cubic functions, with step-by-step calculations for high school calculus students.

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