Which of the following implies that the function has an inflection point at x=c?

Question

There is this multiple choices question in my book and I think three answers are right, the other two are obviously wrong

Those three are

A) $f''(x)$ changes signs at $x=c$

B) f is the stand cubic function and c=0 ( I think this is only true for certain cases, not sure about it. Is the standard cubic function $f(x)=ax^3+bx^2+cx+d$ ? If yes then this one is probably wrong too)

C) $f'(x)$ has a local maximum at $x=c$ (For it to have a local maximum at a point, it's derivative , $f''(x)$ , needs to change signs from posative to negative, making $x=c$ also an inflection point )

Answer 1

We have that

• A is true
• B is not true in general since $f''(x)=6ax+2b\implies f''(0)=2b \neq 0$
• C is true