# Can the product of two functions with no inflection points have an inflection point?

If $$f(x)$$ and $$g(x)$$ are functions with no inflection points, can $$h(x)=f(x)\cdot g(x)$$ have an inflection point?

Edit: I experimented a bit with a few functions (like $$x^2\cdot x^2$$) in a graphing calculator but I couldn't find a good example. I was also running into trouble with the third derivative test in some situations.

I was doing exercises and noticed $$-x^4+4x^3-6x^2$$ doesn't have an inflection point. It can also be rewritten as $$x^2(-x^2+4x-6)$$. This is a product of 2 parabolas and parabolas have no inflection points. If that fact is sufficient to show that the product also doesn't have an inflection point, it would save a lot of time when doing problems like this.

• Think of $x \cdot x^2$. Commented Jun 21, 2021 at 14:48
• Welcome to Mathematics SE. Take a tour. You'll find that simple "Here's the statement of my question, solve it for me" posts will be poorly received. What is better is for you to add context (with an edit): What you understand about the problem, what you've tried so far, etc.; something both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance. Commented Jun 21, 2021 at 14:49
• @user64494 oh, of course. Thank you.
– Milo
Commented Jun 21, 2021 at 14:55
• @MichaelJesurum I see. I thought brevity would be better. Thank you!
– Milo
Commented Jun 21, 2021 at 14:58

Sure! An inflection point means the second derivative is changing sign, so

$$h''(x) = 0$$

at some particular $$x$$, but not in general. To ensure that $$f(x), g(x)$$ have no inflection points, let's just make them quadratic:

$$f(x) = f_2 x^2 + f_1 x + f_0$$

$$g(x) = g_2 x^2 + g_1 x + g_0$$.

Then, expanding around $$x=0$$, we find

$$h''(x) \approx 2 (f_2 g_0 + f_1 g_1 + f_0 g_2) + 6(f_2 g_1 + f_1 g_2) x$$

where we need the first term to vanish but the coefficient of $$x$$ to be nonzero. One possible choice is

$$f(x) = g(x) = x^2 + x -1/2$$.

Plotting verifies visually that indeed $$f(x)g(x)$$ has an inflection point at $$x=0$$ but $$f(x), g(x)$$ do not.