Identify where any horizontal tangents exist for the function:

Question

𝑓(𝑥)=𝑥³+7𝑥²−8

Answer 1

The horizontal tangents occur where the derivative of f(x) equals 0 (f'(x) = 0). So, we can first find the derivative of f(x).

f(x) = x³ + 7x² - 8

Lets find the derivative by applying the power rule:

f'(x) = 3x² + 14x

We can set the derivative to 0 to find the horizontal tangents:

3x² + 14x = 0

x(3x + 14) = 0 Factored out x

x = 0 or 3x + 14 = 0 Solve for x

3x = -14

x = -14/3

So, horizontal tangents exist at x = 0 and x = -14/3.

Answer 2

Horizontal tangents will occur any place the first derivative (slope) is equal to 0, i.e.:

3x² + 14x = 0

Factor and solve for x. You should get two x-coordinates where the slope is equal to 0. You can calculate the associated y coordinates by plugging the x coordinates into the original equation.