# tangent parallel to y or x axis

• Aug 21st 2009, 08:17 AM
skeske1234
tangent parallel to y or x axis
Find the critical points for each of the following. Determine whether the critical point is a local max or min and whether or not the tangent is parallel to the horizontal axis.

y=(-x^2)(e^(-3x))

so i had no problem finding the critical points and max or min points, but I am wondering about whether or not the tangent is parallel to the horizontal axis.
In the back of the book the answer says:

(0,0) is a local max, tangent parallel to the t-axis, (2/3, -4/(9e^2)) is a local min, tangent parallel to t-axis.

Ok, I am thinking that t-axis is a typo and supposed to be y-axis for both cases. BUT, I am not sure why this is true (both parallel to the y-axis).

When I graph it, they both look parallel to the x-axis to me. How do you know if it is parallel to y-axis?
• Aug 21st 2009, 09:01 AM
Haytham
its suppose to be x-axis

if you get any curve object you have lying around (and holding it in any position you like) and use a pencil as the tangent to the point closest to the ground (a local min), that tangent (the pencil) is always horizontal.
• Aug 21st 2009, 09:03 AM
songoku
Hi skeske1234

You find the value of x = 0 and x = 2/3 by setting $\frac{dy}{dx}=0$.

The slope of tangent is equal to $\frac{dy}{dx}$. For x = 0 and x = 2/3, of course $\frac{dy}{dx}=0$. It means that the slope of tangents at both value of x is zero.

If the slope of a line is zero, will it parallel to x-axis or y-axis? :)
• Aug 21st 2009, 09:25 AM
skeske1234
Quote:

Originally Posted by songoku
Hi skeske1234

You find the value of x = 0 and x = 2/3 by setting $\frac{dy}{dx}=0$.

The slope of tangent is equal to $\frac{dy}{dx}$. For x = 0 and x = 2/3, of course $\frac{dy}{dx}=0$. It means that the slope of tangents at both value of x is zero.

If the slope of a line is zero, will it parallel to x-axis or y-axis? :)

Ok, but I thought about it and I think that the (0,0)'s tangent is supposed to be parallel to y-axis and (2/3, -4/(9e^(2))'s tangent is suppposed to be parallel to the x-axis. I think this is tre because when I do the f'(x) test, I note that from x<2/3 (including x<0), f'(x) is negative and x>2/3 is positive.. so, since x<0 and 0<x<2/3 is negative, that means parallel to the y axis or vertical tangent doesnt it?
• Aug 21st 2009, 09:38 AM
songoku
Hi skeske1234

No, f '(x) test helps us to know about the nature of the function. For x < 0, f '(x) is positive. It means that for $-\infty < x < 0$, the function is increasing. For 0 < x < 2/3, f '(x) is negative, so the function on that interval is decreasing. For x > 2/3, f '(x) is positive so the function is increasing.

The tangents at x = 0 and x = 2/3 have the same slope so the tangents will have same orientation. If the tangent at x = 2/3 is parallel to x-axis, which is true, the tangent at x = 2/3 will also parallel to x-axis.