Change in the number of positive zeros of a continuous function.

Question

Let $f(x)$ be any continuous function, then is it true that $$Z^{+}\left(\alpha f(x)+(x+\beta)f'(x)\right)\leq Z^{+}\left(f'(x)\right)+1$$

where $\alpha>1$ is a real number and $\beta$ is any positive integer. $Z^+$ denotes the number of positive zeros.

Note: If $f(x)$ is a polynomial then it's easy to see that the above inequality holds.

Any help or small hint will be really appreciated. Thanks.

Answer 1

Imagine a function with a graph like this:

It is surely smooth, and the flat parts are not quite horizontal, so $f'(x)>0$, and $f(x)<0$, and the steep parts can be made arbitrarily steep. So there are as many roots as we want in the LHS expression, but none in the RHS.

So no, the inequality does not hold.