Calculus - Always true / never true / true in some cases

Hello there, I'm currently struggling with the following assumption:

If the graphs of the functions $f$ and $g$ have a horizontal tangent at the point $x_0$, the graph of $f*g$ also has a horizontal tangent at the point $x_0$.

When I came across this assumption, I visualized it like wave functions interfering. Given the information in the assumption, the waves must be interfering constructively, thus creating a new extremum at $x_0$ , proving the assumption to always be true.

Overlooked the possibility of an inflection point / saddle point. So you have to argue that using the product rule, we obtain $f'(x)=0+0$ for $f*g$ , thus proving the assumption.

#Calculus

Easy Math Editor

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Comments

You can consider the graphs $f(x)=x^2$ and $g(x)=x^3$ , a horizontal tangent is sure to form but it can be an inflection point too!

Jason Gomez - 1 month, 2 weeks ago

Yes, I did not consider that, thank you. Should have just proved it by using the product rule.

Will Schefner - 1 month, 2 weeks ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$