What we can say about the point $x=c$ if $f'(c)=f''(c)=0$ dan $f'''(x) 0$ for all $x$

Question

What we can say about the point $x=c$ if $f'(c)=f''(c)=0$ dan $f'''(x) >0$ for all $x$

agent0smith · Answer

hmm... That it's an inflection point, at least...

agent0smith · Answer

Eg $$\Large f(x) = x^5$$$$\Large f \prime (x) = 5x^4$$$$\Large f \prime \prime (x) = 20 x^3 $$$$\Large f \prime \prime \prime (x) = 60x^2$$
the fourth derivative is positive for all x.
The others are all zero for x=0

watchmath · Answer

why is that true in general?

watchmath · Answer

Sorry actually I wanted to say that $f'''(c)>0$ (so not necessarily $f'''(x)>0$ for all x

agent0smith · Answer

Ok so then it seems like we have an inflection point which is on on upward portion of the graph... like the inflection point of say an x^3 graph vs an -x^3 graph

agent0smith · Answer

I don't think it matters if f''' is > 0 for all x or not... it seems that we just have an inflection point on an upward section of the curve, rather than an inflection point on the downward section of the curve |dw:1381549503654:dw|

or maybe in terms of displacement y...
y' - velocity would be zero 
y'' - acceleration would be zero
but the rate of change of acceleration y''' is positive, as opposed to negative... so acceleration is increasing not decreasing.