Show that the minimum and maximum points of every curve in the family of polynomials
π(π₯) = 2π₯3 + ππ₯2 + 2π₯ lie on the curve π¦ = π₯ β π₯3.
Find the critical number(s)
We consider xβRx\in \RxβR
In this case
Substitute
Then the minimum and maximum points of every curve in the family of polynomials f(x)=2x3+cx2+2xf(x)=2x^3+cx^2+2xf(x)=2x3+cx2+2x lie on the curve y=xβx3.y=x-x^3.y=xβx3.
x=ΜΈ0,cβ€β12x\not=0, c\le-\sqrt{12}xξ =0,cβ€β12β or cβ₯12.c\ge\sqrt{12}.cβ₯12β.
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