let-x-y-z-t-gt-0-and-x-y-z-t-4-prove-that-4-xyzt-2-3-45-x-4-y-4-z-4-t-4-

Tinku Tara June 4, 2023 Algebra 0 Comments

Question Number 150053 by mathdanisur last updated on 09/Aug/21

let x;y;z;t>0 and x+y+z+t=4 prove that (4/((xyzt)^2 )) + 3 ≥ (√(45 + x^4 + y^4 + z^4 + t^4 ))

$$\mathrm{let}\:\:\mathrm{x};\mathrm{y};\mathrm{z};\mathrm{t}>\mathrm{0}\:\:\mathrm{and}\:\:\mathrm{x}+\mathrm{y}+\mathrm{z}+\mathrm{t}=\mathrm{4} \\ $$$$\mathrm{prove}\:\mathrm{that} \\ $$$$\frac{\mathrm{4}}{\left(\mathrm{xyzt}\right)^{\mathrm{2}} }\:+\:\mathrm{3}\:\geqslant\:\sqrt{\mathrm{45}\:+\:\mathrm{x}^{\mathrm{4}} \:+\:\mathrm{y}^{\mathrm{4}} \:+\:\mathrm{z}^{\mathrm{4}} \:+\:\mathrm{t}^{\mathrm{4}} } \\ $$

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