fβ²(x)=6x2+2cx+2 Find the critical number(s)
fβ²(x)=0=>6x2+2cx+2=0
3x2+cx+1=0
x=6βcΒ±c2β12ββ We consider xβR
c2β12β₯0=>cβ€β12β or cβ₯12β In this case
xξ =0,cx2=β3x3βx Substitute
y=2x3β3x3βx+2x
y=βx3+xThen the minimum and maximum points of every curve in the family of polynomials f(x)=2x3+cx2+2x lie on the curve y=xβx3.
xξ =0,cβ€β12β or cβ₯12β.
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