shkusira wrote:

Ian, in statement 2, A can be positive, 0 or negative (Say x=1/2 and A=-1/2).

No, if Statement 2 is true, A cannot be negative. Statement 2 tells us that: Ax^2 + 1 is positive for

all x, and not only for x = 1/2. In your example, where A = -1/2, you can quickly see that Ax^2 + 1 will be negative for x = 10, for example, so A cannot be -1/2; that disagrees with the information in the statement. You can see that, if A is negative, Ax^2 + 1 will be zero (so certainly not positive) whenever \(x = \frac{1}{\sqrt{ |A| }\). Plugging that in for x:

\(Ax^2 + 1 = A \left( \frac{1}{\sqrt{ |A| } \right)^2 + 1 = \frac{A}{|A|} + 1 = -1 + 1 = 0\)

and if \(|x| > \frac{1}{\sqrt{ |A| }\), then Ax^2 + 1 will be negative, if A is negative. So if Statement 2 is true, A cannot be negative. A can, however, be zero, which is why the statement is insufficient.

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