goyalsau wrote:Is A positive?
(1) x^2-2x+A is positive for all x
(2) Ax^2+1 is positive for all x
Here's another approach:
When examining statement 1, we need to answer the following question:
What values of A will guarantee that x^2-2x+A > 0 for all values of x?
The minimum value of x^2-2x is -1, when x = 1. (See footnote below.)
Thus, to guarantee that x^2-2x+A > 0 for all values of x, A must be greater than 1.
Sufficient.
When examining statement 2, we need to answer the following question:
What values of A will guarantee that Ax^2+1> 0 for all values of x?
A=0 works, because then Ax^2+1 = 1, which is positive for all values of x.
A=1 works, because then Ax^2+1 = x^2+1, which is positive for all values of x.
Since we can't tell whether A must be positive, insufficient.
The correct answer is A.
Footnote:
Plug any value other than x=1 into x^2-2x, and you'll get a result bigger than -1. Thus, -1 is the minimum value of x^2-2x. Or to determine what value of x will yield the minimum, set the derivative equal to 0:
2x-2 = 0
x = 1.
Plugging x=1 into x^2-2x, we see that its minimum value is 1^2 - 2*1 = -1.
This latter approach is helpful but beyond the scope of the GMAT.
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