Question

In your lab, a substance's temperature has been observed to follow the function T(x) = (x − 4)3 + 6. The turning point of the graph is where the substance changes from a liquid to a gas. Using complete sentences in your written answer, explain to your fellow scientists how to find the turning point of this function. Hint: The turning point of the graph is similar to the vertex of a quadratic function.

Answer 1

I assume (x-4)3 means (x-4)³.  What we wish is to set the derivative equal to zero.

Expanding the T(x) polynomial makes it easier for me to take the derivative.

So (x-4)³ = x³ - 12x² + 48x - 64 + 6

T'(x) = 3x² - 24x + 48

We can factor out a 3 and set this to zero:

x² - 8x + 16 = 0

(x -4)² = 0

x = 4 should therefore represent the turning point.

I am mildly chagrined, I almost used the f'(x) = nx^(n-1) function at first, which appears would have been correct.