I have got a $4^{th}$ order polynomial, $$x^4+100x^2−2000x+10^4=f(x)$$ whose minimum value I need without using calculus. Is there any way except graph plotting?
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For a start, you can get rid of the huge coefficients by transforming to $y=\frac x{10}$, leading to $f(x)=10^4(y^4+y^2-2y+1)=10^4g(y)$ with $g(y)=y^4+y^2-2y+1$.
If $z$ is the minimum value of $g(y)$, the quartic polynomial $h(y)=g(y)-z=y^4+y^2-2y+1-z$ has a multiple root. A quartic polynomial has multiple roots exactly if its discriminant is zero. The discriminant of $h(y)$ is
$$ -16(16z^3-40z^2+69z-17)\;. $$
Thus the minimum value $z$ is the only real zero of the cubic polynomial $16z^3-40z^2+69z-17$, which is
$$ z=\frac1{12}\left(10-\frac{107}{\sqrt[3]{(174\sqrt{87}-1187}}+\sqrt[3]{(174\sqrt{87}-1187}\right)\approx0.28927\;. $$
Multiply this by $10^4$ to get the minimum value of approximately $2892.7$ of $f(x)$.
joriki
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linear-algebra? – José Carlos Santos Mar 22 '20 at 09:09