Can f''(x_0) = 0 if f'(x_0) =/= 0?

[FaroukYasser]

I was wandering, can $$\ \frac { d^{ 2 }y }{ dx^{ 2 } } | _{ x={ x }_{ 0 } }=\quad 0$$ if $$\frac { dy }{ dx }| _{ x={ x }_{ 0 } }\neq \quad 0$$ and if so, what does this translate to geometrically?

 

[HallsofIvy]

Yes, of course. That is the same as saying that a function can have the value "0" at a point where its derivative is not 0. And it is very easy to give an example of that- suppose y= x- 1. That is a straight line such that y(1)= 1-1= 0 but its derivative is the constant 1 which is never 0. To relate that to your question about first and second derivatives, take an "anti-derivative". An anti-derivative of x- 1 is y= (1/2)x^2- x+ 1. Now, y'= x- 1 which is 0 at x= 1 while the second derivative is 1.

An nth derivative tells how fast the n-1 derivative is changing and is NOT related to the actual value of that n-1 derivative.

 

[Samy_A]

$y=x-x_0$

 

[FaroukYasser]

$y=x-x_0$

Thanks. But I was hoping for a non linear function. Or in other words, I am wandering what this means geometrically. I know that if a point is an inflection point then its second derivative is 0 but the converse doesn't necessarily hold. In other words, what does f''(x_0) = 0 tell us about that point other than it being a likely inflection point.

Thanks :)

 

[FactChecker]

I was wandering, can $$\ \frac { d^{ 2 }y }{ dx^{ 2 } } | _{ x={ x }_{ 0 } }=\quad 0$$

This means that the slope of the curve is not changing at that point

if $$\frac { dy }{ dx }| _{ x={ x }_{ 0 } }\neq \quad 0$$ and if so, what does this translate to geometrically?

This means the slope is not zero at the point. So @Samy_A 's example of a sloped straight line satisfies both conditions at every point.

 

[FactChecker]

In other words, what does f''(x_0) = 0 tell us about that point other than it being a likely inflection point.

Nothing. That is exactly what it means. sin(x) has an inflection point at every x= n * pi. Those are all inflection points with non-zero slopes.

 

[WWGD]

Actually, take a degree-2 or higher polynomial other than $x^n$, i.e. $a_nx^n+...+a_1x+a_0 ; a_j$ not all 0 $n>j\geq 0$, I would say the probability of having both f'(x) and f''(x)=0 is 0 under "reasonable" choices of pdf..

 

[FaroukYasser]

I see. Thanks all!

 

[LCKurtz]

Thanks. But I was hoping for a non linear function. Or in other words, I am wandering what this means geometrically. I know that if a point is an inflection point then its second derivative is 0 but the converse doesn't necessarily hold. In other words, what does f''(x_0) = 0 tell us about that point other than it being a likely inflection point.

Thanks :)

The second derivative geometrically identifies concavity. f'' > 0 or f''<0 on an interval identifies concave up or down, respectively. In a case where f'' is continuous, the only way for concavity to switch is for f'' to pass through zero. So f'' = 0 at a point might be a case where f'' is changing sign, indicating an inflection point (change of concavity). Of course, f'' might not change sign nearby so it might not be an inflection point. I wouldn't say such a point is a "likely" inflection point but a "possible" inflection point.