Second derivative =0 (1 Viewer)

[chocolatelatte]

okay so I am aware that when f"(x)=0, it can be an inflexion point, max, min, anything really..... am I right?
WHY THEN, IS F'(2)=F"(2)=0 A HORIZONTAL POINT OF INFLEXION? can't it also be a turning point?

I AM SO CONFUSED.
thank you !!

 

[rumbleroar]

first of all, at f"(x) = 0 a inflexion point may not occur (re: case of x^4) and you need to test it to confirm whether or not it is an inflexion point. and no it cannot be a max or min. max and min relate to the turning points or stationary points, i.e. when f'(x) = 0

f'(2)=f"(2)=0 is a HPOI (horizontal point of inflexion), not a turning point because the curve will be continually increasing or decreasing, its concavity doesn't change. if its concavity changed on either side of x, then its a turning point.

iirc, HPOI occurs when the stationary point (i.e. f'(x)=0) is neither a minimum nor maximum, but has its concavity change when f"(x)=0

I hope that clarifies things for you

 

[funnytomato]

okay so I am aware that when f"(x)=0, it can be an inflexion point, max, min, anything really..... am I right?
WHY THEN, IS F'(2)=F"(2)=0 A HORIZONTAL POINT OF INFLEXION? can't it also be a turning point?

I AM SO CONFUSED.
thank you !!

F'(2)=F"(2)=0 does not necessary imply A HORIZONTAL POINT OF INFLEXION

You need to verify there's a change of concavity
i.e. F''(x) has opposite signs to either side of x=2

e.g. for f(x)= x^3
f''(0)=f'(0)=0
ALSO, f''(x)>0 for x>0 , f''(x)<0 for x<0 ====> f'' has opposite signs, concavity changes
Hence (0,0) is a horizontal point of inflexion

Whereas in your case , you can just for example take F(x)=(x-2)^4
then the condition F'(2)=F"(2)=0 is satisfied
but F''(x)=12*(x-2)^2 > 0 to either side of x=2 ====> F'' has the same sign, concavity DOES NOT change
So the point (2,0) is NOT a horizontal point of inflexion
And it is actually a minimum by finding the signs of F'(x)

Anyways, what I want to say is that :

You ALWAYS need to check the change in sign of f'' (hence change of concavity) for a point of inflexion