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To every parameter $ x \in ( - \infty , 1]$ let $f(x)$ be the biggest real zero of $ p_x (z) :=z^4 - 4z^3 + 6z^2 - 4z+x$.

How can I determine $ f(x) $ by substitute $z=u+1$ und calculate the relative condition of $f$ at $x=1$?

Would be very helpful if someone can explain to me what this is all about. Thanks in advance

Jean Marie
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wondering1123
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    Notice that $z^4-4z^3+6z^2-4x+1 = (z-1)^4$, by looking at the coefficients, and comparing them with the binomial expansion. Therefore, you can write it as $(z-1)^4 + (u-1)$. How can you express the zeroes of the function now? – Toby Mak Oct 24 '17 at 09:20

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Note $p_x(z) = (z-1)^4+(x-1)$. Thus substitution $z=u+1$, gives $u^4=1-x$ as the equation to seek roots for, whence $u = \sqrt[4]{1-x}$ is clearly the largest (real) root and hence $f(x) = 1+\sqrt[4]{1-x}$.

Macavity
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