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Determine the values of the real parameter $m$, so that the equation $x^4-2x^3+2mx-m^2=0$ admits four different real solutions.

$$x^4-m^2+2mx-2x^3=(x^2+m)(x^2-m)-2x(x^2-m)=(x^2-m)(x^2-2x+m)$$ From the first factor $m\gt0$ From the second factor do determinants greater than or equal to 0 and so the solution set will become $$m\...
Yash Shrivastava's user avatar
2 votes
Accepted

Determine the values of the real parameter $m$, so that the equation $x^4-2x^3+2mx-m^2=0$ admits four different real solutions.

You can continue with your idea. To get $4$ solutions in $x$ you need both $x^2=m$ and $x(2-x)=m$ to provide $2$ solutions each. The first one requires $m>0$, and the second $m<1$, so you know ...
zwim's user avatar
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2 votes
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Parameters behind non-symmetric Lissajous loop?

To rephrase your question: find two functions of $\theta$, $f,g$ such that the trace $[x,y]=[f(\theta),g(\theta)]$ traces out your curve. The quick and dirty solution is to read off the functions $f,g$...
user619894's user avatar
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2 votes
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Determine the length of the rod that can be inscribed in a cuboid

This problem can be solved easily with a minimization procedure. By symmetry, the rod central axis should pass by the point $q_0 = (a,b,c)$. Now let be $$ \matrix{ p_0 = (x_0,y_0,z_0) & \text{...
Cesareo's user avatar
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1 vote

Is it "ok" to see constants as a "section" in $n$ dimentional space to get to lower dimentionality space?

In highschool mathematics you learn that there are "variables", such as $x, y, z, t$, "parameters", such as $a, b, p, q$ and "constants" such as $\pi$, $e$ and $\sqrt{2}$....
M. Wind's user avatar
  • 3,713

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