Any Theorems on Number of Roots of a Polynomials of Decimal Exponents?

Question

I want to prove the number of maximum possible roots for the function $f$ mathematically. Are there any theorems or properties that I can use?

$f(x)=-2x^{0.4}+5(\frac{3}{34}x^{1.7}-\frac{1}{42}x^{2.1})$

Answer 1

Let $x=y^{10}$. We have

$$-2y^4+5\left(\frac3{34}y^{17}-\frac1{42}y^{21}\right)=0.$$

By the fundamental theorem of algebra, the number of roots is bounded by $21$.

As we can factor

$$x^{4/10}\left(-2+5\left(\frac3{34}y^{13}-\frac1{42}y^{17}\right)\right)=0.$$

one root is $x=0$ and there are no more than $17$ others.

Taking the derivative of the second factor, we can show that there are exactly two real roots for $x>0$.

Answer 2

Write $x=e^y$, and $g(y)=f(e^x)$. Assume $g$ has three roots (with multiplicity). Then for some scalar $c$, $(e^{-2.1y}g(y))-c$ has three roots with multiplicity.

Therefore, by Rolle, $(e^{-2.1y}g(y))’=\alpha e^{0.4y}+\beta e^{1.7y}=h(y)$ has two roots.

Trouble is that if $h(y)=0$ then $e^{1.3y}=-\frac{\alpha}{\beta}$, so $y$ is uniquely determined so $h$ has at most one root (with muliplicity $1$).