How can I explain that points are inflection points and not critical points. I know the inflections points are where the concavity changes, but not sure how to say why a point is not a critical point.
Does this make sense?

Question

amistre64 · Answer

a critical point tends to be the top of a hill or the bottom of a valley.  inflections are just half markers...

amistre64 · Answer

sometimes an inflection point is disguised as a critical point because the slope at the inflection is zero.....

anonymous · Answer

I had worded it as the critical points tell max and mins and the inflections points shows where concavity changes.

amistre64 · Answer

that makes sense to me.... :)

amistre64 · Answer

all points are "critical" when it comes to defining the curve...but when we want to maximise or minimise the results of our inputs, we search for the "critical" points of the curve to give us our results....

anonymous · Answer

thanks
my problem was x^4 + 6x^3 - 24x^2 + 26

Show 2nd derivative.
show inflections point(s) and explain in complete sentences why these numbers are inflection points and not critical points.

I got 2nd derivative as 12x^2 + 36x - 48
with inflection points as x=1  x=-4.

anonymous · Answer

if I did this correctly. When I graph it the 1 & -4 are not at the top or in the valley

amistre64 · Answer

its a quartic graph with a positive slope which means that it has at most 2 inflection points.  if not "only" 2 inflection points :)

so I agree with your assessment..
x^2 +3x -4

(x+4)(x-1)  x = -4,1