Find k such that the line is tangent to the graph of the function. Function:f(x) = kx^3 Line:y = 7x + 2

Question

Find k such that the line is tangent to the graph of the function. Function:f(x) = kx^3 Line:y = 7x + 2

Answer 1

The key to solving this problem is to understand what is needed in order to find a tangent line. A tangent line requires a slope and a point. The problem has given you the slope; you can find it in the equation of the line. Where does the tangent slope come from? The derivative of the function gives you the slope at any point on that function. Once you have identified the slope from the line and the derivative from the function, they will be equal to each other. Because you are trying to find k, solve for x in this new equation.

Now, for a line to be tangent to a function, that line needs to intersect the function at only one point. When two functions intersect, they are equal. Now set the function equal to the line. At this point, there are two variables and that is not helpful. Use your work from the paragraph above to make the substitution and solve for k. (This part could be messy, and the answer will work out to be something quite simple.)

Answer 2

f(x) = kx³

f'(x) = 3kx²

3kx² = 7

3kx² = 7

k = 7 / 3x²