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.
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.
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?
You find the value of x = 0 and x = 2/3 by setting .
The slope of tangent is equal to . For x = 0 and x = 2/3, of course . 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?
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 , 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.