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2007/08 AHSMC Part 1 Problem 16

Problem 16 of 16

The smallest value of the real number kk such that (x2+y2+z2)2≤k(x4+y4+z4)(x^2 + y^2 + z^2)^2 \le k(x^4 + y^4 + z^4) holds for all real numbers xx, yy and zz is

Answer choices

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Solution

We can rewrite the inequality as (k−3)(x4+y4+z4)+(y2−z2)2+(z2−x2)2+(x2−y2)2≥0(k-3)(x^4 + y^4 + z^4) + (y^2-z^2)^2 + (z^2-x^2)^2 + (x^2-y^2)^2 \ge 0, from which it is clear that k≥3k \ge 3. The answer is (C).

University of Alberta