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.

  • $\begingroup$ Why is this tagged multivariable calculus, geometry, and algebraic geometry? Where did you encounter this problem? $\endgroup$ – apnorton Feb 20 '13 at 21:12

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 $y=0$.

  • $\begingroup$ Ok. I guess I should limit my statement to non-trivial expressions... $\endgroup$ – apnorton Feb 20 '13 at 21:28
  • $\begingroup$ Would it be possible to find a range of c values for y such that f(x, c) = 0 for all a <= x <= b, instead of just for a single value of c? $\endgroup$ – user1855952 Feb 20 '13 at 21:34
  • $\begingroup$ @user1855952 So... for all $c \in [\alpha, \beta]$, there exists a $d \in[a, b]$ such that $f(d, c) = 0$? This is what you're asking? $\endgroup$ – apnorton Feb 20 '13 at 21:40
  • $\begingroup$ yup, is this a solvable problem? $\endgroup$ – user1855952 Feb 20 '13 at 21:43
  • $\begingroup$ @user1855952 Not necessarily. How about the paraboloid $$f(x, y) = x^2 + y^2 + 5$$. No matter how hard you try, you will find no interval $[\alpha, \beta]$ that satisfies the above, as it never intersects the xy plane. $\endgroup$ – apnorton Feb 20 '13 at 21:49

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