Brief Calculus.

Question

The graph of  has two horizontal tangents. One occurs at a negative value of  and the other at a positive value of . What is the negative value of  where a horizontal tangent occurs?

x=

What is the positive value of  where a horizontal tangent occurs?

x=

Answer 1

We can use the power rule to take the first derivative of the function, set it = 0, and solve using quadratic formula or factoring (since the first derivative of a cubic function will be quadratic):

f^'(x) = 6x² + 6x - 120 = 0 (since f^' gives us the slopes of tangents to f, and the slope of a horizontal line = 0)

x² + x - 20 = 0 (dividing both sides by 6)

(x - 4) (x + 5) = 0

x = 4 or x = -5