[latex]f)\ x^4-x^2+24 geq 12(x^2-1)\ x^4-x^2+24 geq 12x^2-12\ x^4-x^2+24 -12x^2+12 geq 0\ x^4-13x^2+36 geq 0\ \ Niech x^2=t wedge t geq 0\ \ t^2-13t+36 geq 0\ Delta=169-144=25\ sqrt{Delta}=5\ \ t_1=frac{13-5}{2}=4\ t_2=9\ \ x^2 = 4 vee x^2=9\ x= -2 vee x=2 vee x=-3 vee x=3\ \ oxed{xin(-infty;-3 extgreater cup extless -2,2 extgreater cup extless 3,+infty)}[/latex] [latex]d)\ x^4 leq 8x^2\ x^4-8x^2 leq 0\ \ Niech x^2=t wedge t geq 0\ \ t^2-t leq 0\ t(t-1) leq 0\ t_1=0 vee t_2=1\ \ x^2=0 vee x^2=1\ x=0 vee x=-1 vee x=1\ \ oxed{xin(-infty;-1 extgreater cup extless 0,1 extgreater }[/latex]
d) [latex] x^{4} leq 8 x^{2} [/latex] [latex] x^{4}-8 x^{2} leq 0[/latex] [latex] x^{2} =t[/latex] i [latex]t geq 0[/latex] [latex]t^{2} -t leq 0[/latex] [latex]t(t-1) leq 0[/latex] t=0 lub t=1 [latex] x^{2} =0[/latex] lub [latex] x^{2} =1[/latex] x=0 lub x=1 lub x=-1 x ∈(-∞,-1> ∪ <0,1> f) [latex] x^{4} - x^{2} +24 geq 12( x^{2} -1)[/latex] [latex]x^{4} - x^{2} +24 geq 12x^{2} -12[/latex] [latex]x^{4} - x^{2} +24-12 x^{2} +12 geq 0[/latex] [latex]x^{4} - 13 x^{2} +36 geq 0 [/latex] [latex] x^{2} =t[/latex] i [latex]t geq 0[/latex] t²-13t+36≥0 Δ=169-4*36=169-144=25 √Δ=5 [latex] t_{1} = frac{13-5}{2} =4[/latex] [latex] t_{2} = frac{13+5}{2} =9[/latex] [latex] x^{2} =4[/latex] lub [latex] x^{2} =9[/latex] x=2 lub x=-2 lub x=3 lub x=-3 x∈(-∞,-3> ∪ <-2,2> ∪ <3,+∞)
