Tangent line using basic differentation

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?   

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

Answer 1

A horizontal line has a slope of zero. Since the derivative is a function of the slope of the line tangent to the original function, we just need to find the derivative and set it equal to zero.

For f(x) = 2x³ + 9x² - 60x + 13, using the power rule, the derivative is f '(x) = 6x² + 18x - 60 so:

6x² + 18x - 60 = 0

x² + 3x - 10 = 0

(x + 5)(x - 2) = 0

x = -5, x = 2

So at x = -5 and x = 2, the slope of the tangent line is zero