2022 · Q11
Consider the following statements :
- 1x⁴(y - z) + y⁴(z - x) + z⁴(x - y) is positive if x > y > z.2. x⁴(y - z) + y⁴(z - x) + z⁴(x - y) is negative if x < y < z.
Which of the above statements is/are correct?
Elementary MathematicsAlgebra
- AOnly 1
- BOnly 2
- CBoth 1 and 2
- DNeither 1 nor 2
Explanation
The expression is antisymmetric of degree 5 and equals −(x−y)(y−z)(z−x)(x²+y²+z²+xy+yz+zx) (verified at x=3,y=2,z=1, where it is 50). The symmetric factor is always positive, so the sign is opposite to that of (x−y)(y−z)(z−x): positive when x>y>z and negative when x<y<z. Both statements are correct. The keyed option is C.