Is the given statement is true or false? Justify your answer.

Question

If f"(c) = 0, then (c, f(c)) is an inflection point of the curve

y = f(x)

Answer 1

y = x^3 has an inflection point at (0,0)

It's where the concavity changes from up to down or down to up

y' = 3x^2 = 0 when x = 0

y" = 6x = 0 when x = 0

then y=x^4 is a W or UU shaped curve

then y'=4x^3

and y" = 12x^2

still when x = 0, y = 0

while x=0 doesn't change concavity from up to down, for an instant, an infinitesmal second, it almost does. and back.

so conclusion? Almost. in a surreal number system yes. but for real numbers no. if you're into reality, no.

Answer 2

False. Consider y=x^4 at (0, 0)