Consider the following statements:<br><br>1. x^{4}(y-z)+y^{4}(z-x)+z^{4}(x-y) is positive if x \gt y \gt z.<br>2. x^{4}(y-z)+y^{4}(z-x)+z^{4}(x-y) is negative if x \lt y \lt z.<br><br>Which of the above statements is/are correct?
- A. Only 1
- B. Only 2
- C. Both 1 and 2 ✓
- D. Neither 1 nor 2
Correct Answer: C. Both 1 and 2
Explanation
The expression factors as -(x-y)(y-z)(z-x)(x^2+y^2+z^2+xy+yz+zx). If x \gt y \gt z, the product (x-y)(y-z)(z-x) is negative, making the whole expression positive. If x \lt y \lt z, the product is positive, making the expression negative. Thus, both statements hold true.
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