Not necessarily. I always mean to write this up systematically.
1) For a "standard" x-y graph, If
dy/dx=0 then the gradient at that value of x is horizontal which you identify as a maximum , minimum or point of inflection. More accurately, the graph could be horizontal everywhere e.g. y = 2. A point of inflection can be sloping e.g. tan x for x = 0. Another example, between the maximum and minimum of a simple cubic, y = (x+1)(x-1)(x+2), the (sloping) point of inflection occurs where the derivative has a maximum or minimum (that's where the second derivative comes in).
For y =
x3 at x= 0 then dy/dx=0 and
d2y/dx2 = 0 and we
do have P.O.I at x=0.
Problem occurs for y=
x4. Here dy/dx
and d2y/dx2 are
zero for x=0 but
NOT a P.O.I.
I think an argument goes like this.
Consider y =
x2(x+a)2. For a = 0 the
first derivative has an
odd number of coincident (repeated) roots at x= 0 , the second derivative
is zero but there is
NOT a P.O.I at x=0. For a
=0 you can't create an even number of coincident roots for first derivative so can't create horizontal P.O.I for this function.
Consider y =
x3(x+a) . For a
=0 the
first derivative has an
even number of repeated roots at x = 0, and there is a
horizontal P.O.I at x=0. As a approaches zero the derivative gets an
odd number of repeated roots at x=0 and P.O.I becomes a minimum (no longer a P.O.I even though second derivative is zero here (at x = 0)).
I am sure I can write that more clearly but I think the idea is (mostly) correct. I am sure someone will put me right if not. Whatever, this is why the textbooks avoid discussing nature of turning points when first and second derivative are zero for same value of x.
The "old boys" were taught a rule of repeated differentiation (fourth, 5th, etc derivatives) to determine the "real nature" of every turning point and the location of sloping P.O.I.s.