Is A positive?
1. x^2 - 2*x + A is positive for all x
2. A*x^2 + 1 is positive for all x
took a long while for me to get my head around this...very tricky question! From statement 1
x^2 - 2*x + A > 0, for all x-values
[highlight](x-1)^2[/highlight] + A - 1>0
For all values of x, highlighted portion will always be positive.
In order for whole equation to hold true for all values of x, A has to be at least 1 i.e. A>0(as this will render A-1 to be zero)
If A were to be zero or negative, it will not suffice the equation(as it will yield positive values only for some values of x).
OE is: St1. can be rewritten as (x - 1)^2 + A – 1 > 0, for this to hold true for all possible values of x, A > 1. So sufficient. From statement 2
Ax^2 + 1 > 0, for all x values
for this statement to stand true, A must be more than or equals to zero (i.e. not necessarily positive).
If A were to be negative(i.e. <0), the statement may not stand true for some values of x.
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