# Show that #1<=(1+x^3)^(1/2)<=1+x^3# for #x>=0# ?

See below.

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There are probably more elegant solutions, but here are two.

The result follows.

Another approach

Undoing the substitution, we get

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To show ( 1 \leq (1+x^3)^{1/2} \leq 1+x^3 ) for ( x \geq 0 ), we can square both sides of the inequality ( 1 \leq (1+x^3)^{1/2} ). This gives us ( 1 \leq 1+x^3 ). This inequality is always true for ( x \geq 0 ).

Next, we need to show that ( (1+x^3)^{1/2} \leq 1+x^3 ) for ( x \geq 0 ). Squaring both sides of the inequality ( (1+x^3)^{1/2} \leq 1+x^3 ) gives us ( 1+x^3 \leq 1+2x^3+x^6 ). Simplifying this further, we get ( 0 \leq x^3(x^3+2) ), which is true for ( x \geq 0 ).

Therefore, we have shown that ( 1 \leq (1+x^3)^{1/2} \leq 1+x^3 ) for ( x \geq 0 ).

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