So let's say I have some random equation $6yx^3 - 3yx + 5 = 0$, but could be anything.
How would I go about finding a value for $y$, that makes it so that this equation only holds true if $x$ is some value within the range $[a, b\textbf{]}$.
Also, is it possible to solve such a problem using wolfram alpha? If so, how would I input it.
We have the expression $f(x,y)=0$. There is no value $y=c$ such that $f(x, c) = 0$ for all $x \in [a, b]$.
Why? We would be looking for a $c$ such that the polynomial $f(x,c)$ (I'm assuming this is a polynomial based on the question tags) has infinitely many solutions, namely those in $[a, b]$. This violates the fundamental theorem of algebra.
EDIT: As noted in the comments, this isn't valid for the null polynomial. Namely, if your expression is $f(x, y) = x^2y + xy$, $f(x, y)$ has infinitely many solutions for the choice $y=0$.