can an asymptote be an inflection pt?

Answer 1

An asymptote itself can't be an inflection point. However, there are some graphs which are defined at an asymptote (early in the graph, as opposed to at x goes to -infinity or infinity. I can't think of an example right now, but there HAVE been graphs that initially go through an asymptote and ultimately try and reach it once again.

So in that sense, the graph at which the asymptote occurs *can* be an inflection point. Once I find that example, I'll edit it here. The example I have written down here has these characteristics:

(1) f(x) decreasing from (-infinity, -2]
(2) f(x) is increasing from [2, infinity)
(3) lim (x -> -infinity) = 0
(4) lim (x -> infinity) = infinity
(5) f(x) is concave down from (-infinity, 4) U (0, infinity), and
concave up from (-4, 0)
(6) Local minimum at (-2,3)
(7) f(0) = 0

(0,0) is an inflection point, and y = 0 is an asymptote.
Asymptote at 0 but still defined at 0.

Answer 2

Looks like asymptote is covered.
An inflection point is that point at which the second derivative changes sign. Graphically, it's where the curvature changes to the opposite direction. Think of the graph of x^3. The inflection point is at x=0, where you can see the curvature changes to the opposite direction. Also sometimes of interest is the slope (first derivative) at the point of inflection. For x^3, the slope (x^2) at the inflection point is zero. The inflection points of the sin function (1st derivative = cos(x), 2nd derivative = sin(x)) occur every 180 deg (or every pi radians) and the slope at the inflection points alternate +1 and -1.
So as you can see, a function must pass through the inflection point of necessity, while a function never attains the value of an asymptote in the limit function (this however does not preclude the function from attaining the value of the asymptote in some other region)

Answer 3

Asymptote, a straight line associated with a curve, having the property that as a point moves along the curve to infinity, the distance from the point to the straight line tends toward zero. Some definitions of an asymptote require that after a certain distance along the line, the curve and the line do not intersect. In figure 1, the lines x = 0 and y = x are vertical and slant asymptotes, respectively, to the graph of y = x + 1/ x. The x and y axes are horizontal and vertical asymptotes to the graph of the hyperbolay = 1/x.

Horizontal asymptotes can sometimes appear in population growth graphs when the growth of the population is inhibited by some factor, such as a limited amount of food. This type of growth is often modeled by the logistic growth function

where P(t) is the size of the population at time t. As the graph of P(t) in figure 2 shows, the population continues to grow closer and closer to the maximum number L without ever reaching it. The line y = L is a horizontal asymptote to the graph of P(t).

Asymptotes can be defined formally using the idea of limits in calculus. The line y = L is said to be a horizontal asymptote to the graph of the function y = f(x) if the limit of f( x) as x tends toward infinity or toward minus infinity is equal to L. The line x = L is said to be a vertical asymptote if the limit of f(x) as x approaches L (from either the right or the left of L) is equal to either infinity or minus infinity.

Answer 4

No because the functions would never actually "touch" an asymptote they would only get close to it, but for a point to be an inflection point the function has to exist at that point.

Answer 5

An inflection point only applies when the function is defined so an asymptote cannot be an inflection point (since the function is not defined there).

Answer 6

i think that asymptote is a line.
so it never can be a inflection point.

but if you have a function that blablabla at a certain time haves a asymptote, that may means a inflection point for that function...