calculus help me

Question

Let f(x) = x^3 − 9x^2 − 16x + 1 and g(x) = x^3 − 0.5x^2 + x. Find the value of x where the tangent line of the graph of f(x) is parallel to the tangent line of the graph of g(x)

Answer 1

The first derivatives are equal at that point

3x² - 18x - 16 = 3x² - x + 1

Can you solve for x and answer?

Answer 2

Hello Irakli! Recall that the derivative of a function will give you another function that can give you the slope at any point of the original function. If the tangent lines of f(x) and g(x) are parallel to each other then the slopes of those tangent lines are equal. Therefore if we take the derivatives of both functions and set them equal to each other we can find the unique value of x that will make both tangent lines parallel. In some cases there can be more than one value of x but not in this case.

so f"(x) =g"(x)

Using the power rule for each term we obtain the two derivatives and set them equal to each other:

3x² -18x -16 = 3x² - x +1

Solving for x we get x = -1

This is your anwer but lets check it. You are not asked for the actual tangent lines so you can stop here. The slopes for two tanget lines should be equal. Substituting -1 into both derivaties gives a slope of 5 for both tangent lines.

f"(-1) = g"(-1) = 5