# How do you determine at which the graph of the function y=1/x^2 has a horizontal tangent line?

Mar 13, 2018

By using derivatives

#### Explanation:

derivatives define the slope of a tangent line at a point on the function
therefore if the tangent line is horizontal, its slope is 0

so, on differentiating
$y ' \left(x\right) = \frac{d}{\mathrm{dx}} \frac{1}{x} ^ 2 = 0$

we're setting it equal to zero because we want to see the points at which the derivative is 0 so its slope Is 0

we can use the power rule here as $\frac{1}{x} ^ 2$ is just ${x}^{-} 2$

therefore,

$- 2 {x}^{-} 3 = 0$
divide both sides by $- 2$

$= \frac{1}{x} ^ 3 = 0$

this is an indeterminate form as the only way this could satisfy the equation is if x was positive or negative infinity
therefore at finite values of x, we don't ever have a point where the tangent lines are horizontal

you can see this on the graph that as x becomes bigger and bigger its slope decreases and gets closer and closer to 0, so as x approaches infinity, its slope approaches 0

graph{1/x^2 [-12.66, 12.65, -6.33, 6.33]}