Can we define inflection point to be the relative extremum of $f'$?

Question

In calculus, the inflection point of a function are treated either informally, or putting more attention on its intuition than precise definition. By making some observation, I come up with an idea that it seems that the inflection point can defined as the relative extremum of $f'$. I wonder if it's OK and make no difference with the intuitive meaning as everybody knows. And I also wonder if there are other definitions used by some people.

Answer 1

Even just being a critical point of $f'$ does not guarantee an inflection point, which you can see with the function $$ f(x)=cx^2 $$ $f'(x)=2cx$, $c\ne 0$ which is zero at the origin, but $f''(0)=2c\ne 0$ and so concavity does not change at the origin, as you would expect.

Answer 2

Assuming a smooth curve

The curve $y=\mathrm f(x)$ has an inflexion when $\mathrm f''(x)=0$. The inflexion is called "ordinary" if $\mathrm f'''(x) \neq 0$ and "higher" if $\mathrm f'''(x)=0$. Some elementary texts require $\mathrm f'(x)=0$, but this is just an inflexion whose tangent line is horizontal. The curve $y=\sin x$ has an inflexion at $(x,y)=(0,0)$.

An inflexion point is a point where a curve has unusually high-order contact with its tangent line.

A curve and its tangent line usually have order-2 contact, i.e. the simultaneous solution of the curve's equation and the tangent line's equation has a double root.

For example: Consider the curve $y=x^2$ and the tangent line $y=2x-1$ at $(x,y)=(1,1)$. Solving $y=x^2$ and $y=2x-1$ simultaneously gives $$2x-1=x^2 \iff x^2-2x+1=0 \iff (x-1)^2=0$$ This has a double root at $x=1$ and so the tangent line and curve have order-2 contact.

For example: The curve $y=x(x^2-1)$ has an inflexion at $(x,y)=(0,0)$. To see this, find the equation of the tangent line there: $y=-x$. Solving $y=x(x^2-1)$ and $y=-x$ simultaneously: $$-x = x(x^2-1) \iff x^3=0$$ which has a triple root at $x=0$. So the curve and tangent line have order-3 contact at $(x,y)=(0,0)$ and that point is an inflexion.

For example: The curve $y=\sin x$ has an inflexion at $(x,y)=(0,0)$. The tangent line is $y=x$ so we need the order of the root of $x-\sin x =0$. When $x=0$, clearly $0-\sin 0 =0$. We can differentiate to give $1-\cos x =0$, and this has a solution when $x=0$. We differentiate again to give $\sin x =0$ which has $x=0$ as a solution. Differentiating again gives $\cos x = 0$ which is false when $x=0$. This tells us that $x - \sin x =0$ has a triple root when $x=0$. (The equation $\mathrm g(x)=0$ has a root of order-$k$ if $\mathrm g(x)=\mathrm g'(x)=\cdots =\mathrm g^{(k-1)}(x)=0$ and $\mathrm g^{(k)}(x) \neq 0$. )