mohaprasad wrote:If x and y are nonzero integers, is x^y < y^x ?
(1) x = y^2
(2) y > 2
Need help with the above problem.
Statement 1: x = y²
Substituting y=x² into the question stem, we get:
(y²)^y < y^y²
y^(2y) < y^y².
Question stem, rephrased:
Is y^(2y) < y^y²?
Test one case that also satisfies statement 2.
Case 1: y=3
Plugging y=3 into the y^(2y) < y^y², we get:
3� < 3�.
YES.
Test one case that doesn't also satisfy statement 2.
Case 2: y=1
Plugging y=1 into y^(2y) < y^y², we get:
1² < 1¹
NO.
Since the answer is YES in Case 1 but NO in case 2, insufficient.
Statement 2:
Case 1 also satisfies statement 2.
In Case 1, the answer to the question stem is YES.
Case 3: y=4, x=-1
Plugging x=-1 and y=4 into x^y < y^x, we get:
(-1)� < 4^(-1)
1 < 1/4.
NO.
Since the answer to the question stem is YES in Case 1 but NO in Case 3, INSUFFICIENT.
Statements combined:
Case 1 satisfies both statements.
In Case 1, the answer to the question stem is YES.
Test an extreme case that satisfies both statements.
Case 4: y=10
Plugging y=10 into y^(2y) < y^y², we get:
10²� < 10¹��.
YES.
Case 4 illustrates that -- when both statements are satisfied -- the lefthand side will always be less than the righthand side.
SUFFICIENT.
The correct answer is
C.
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