# find all values of x (if any)

#### Islandguy

##### New member
Find all values of
x
(if any) where the tangent line to the graph of the given equation is horizontal.

y = −9x^28x

#### MarkFL

##### Super Moderator
Staff member
Hello, and welcome to FMH!

How do we go about finding the slope of the line tangent to a given curve? What is the slope of a horizontal line?

#### Islandguy

##### New member
slope of horizontal line is 0

y=mx+b

#### MarkFL

##### Super Moderator
Staff member
slope of horizontal line is 0
Yes, so how can we find the slope of the line tangent to the given function?

#### Islandguy

##### New member
I found the derivative which is -18×-8 but I am stuck here

#### MarkFL

##### Super Moderator
Staff member
I found the derivative which is -18×-8 but I am stuck here
Yes:

$$\displaystyle y'=-18x-8$$

This tells us that for a given value of $$x$$, what the slope of the line tangent to the curve at $$(x,y)$$ will be. We want this slope to be zero, because the tangent line is to be horizontal. So, equate the derivative to zero, and solve for $$x$$...what do you get?

-10?

#### Otis

##### Senior Member
Hello Islandguy. That y is not the same as the y in your exercise, so I would pick a different symbol when writing the slope-intercept form for an arbitrary tangent line to the curve in this exercise.

Y = mx + b

But, we're not asked to report an equation for the horizontal tangent line, so actually we don't need the slope-intercept form at all. Let's just go with m=slope.

As we move along the given parabola in this exercise, x changes. As x changes, so does m. (There are lots of different tangent lines along the parabola, and each has its own slope.)

Did your calculus class mention the following?

The first derivative is a slope. That is, at each point on the curve of a function y, the value of m is y'.

In other words, the first derivative of a function gives us the tangent line's slope (at each x in the domain). In this regard, the first derivative is a function of x itself: a function of slopes.

When we're interested in where a tangent-line slope is zero, then we're interested in where the first derivative is zero.

#### MarkFL

##### Super Moderator
Staff member
We want to solve:

$$\displaystyle -18x-8=0$$