Finding inflection points

Well, I am currently employing myself in finding the inflection points to the function y = 4x^2/(4-x^2). Through the process, the obtain the second derivative

y" = 32(4+3x^2)/(4-x^2)^3. I know from the definition of inflection points that x = -2 and x = 2 do not fit the criterion of being inflection points, and there are no real zeros; yet clearly, from the graph, I can see that there are changes in concavity. So are the two value, x = 2 and x = -2, pseudo points of inflection?

Re: Finding inflection points

Point of inflection will occur when roots are repeated three times.

I.e. $\displaystyle y = x^3$ has a point of inflection at $\displaystyle x=0$

$\displaystyle y = (x-a)^3$ has a point of inflection at $\displaystyle x=a$

$\displaystyle y = x(x-a)(x-b)^3$ has a point of inflection at $\displaystyle x=b$

and so on...

Re: Finding inflection points

Quote:

Originally Posted by

**Bashyboy** Well, I am currently employing myself in finding the inflection points to the function y = 4x^2/(4-x^2). Through the process, the obtain the second derivative

y" = 32(4+3x^2)/(4-x^2)^3. I know from the definition of inflection points that x = -2 and x = 2 do not fit the criterion of being inflection points, and there are no real zeros; yet clearly, from the graph, I can see that there are changes in concavity. So are the two value, x = 2 and x = -2, pseudo points of inflection?

the curve is concave down on $\displaystyle (-\infty,-2)$ and $\displaystyle (2,\infty)$

the curve is concave up on $\displaystyle (-2,2)$

note that y is undefined at $\displaystyle x = \pm 2$ ... a curve cannot have inflection points where it is not defined. **Inflection points are points defined on a curve** where y'' changes sign.