Is A positive?
1. x^2-2x+A is positive for all x
2. Ax^2+1 is positive for all x
Alternate approach:
Statement 1:
In other words, the graph of y = x² - 2x + A lies entirely above the x-axis, so that the value of y is always positive.
A parabola of the form y = ax² + bx + c, where a>0, opens UPWARD.
The result is a U-shaped graph that looks like this:
U.
The DISCRIMINANT of the parabola is equal to b² - 4ac.
The U-shaped graph will lie entirely above the x-axis -- and thus will yield only positive values for y -- if its discriminant is negative.
Implication:
Since y = x² - 2x + A must yield only positive values for y, its discriminant must be negative.
In y = x² - 2x + A, a=1, b=-2, and c=A.
Since b² - 4ac < 0, we get:
(-2)² - 4*1*A < 0
4 - 4A < 0
-4A < -4
A > 1.
SUFFICIENT.
Statement 2:
In other words, the graph of y = Ax² + 1 lies entirely above the x-axis, so that the value of y is always positive.
Case 1: A=0, so that y = Ax² + 1 becomes y=1.
Here, the graph is a horizontal line that lie entirely above the x-axis.
Case 2: A=1, so that y = Ax² + 1 becomes y = x² + 1
Here, since x² cannot be negative, every value for y will be positive, yielding a graph that lies entirely above the x-axis.
Since A=0 in Case 1 but A>0 in Case 2, INSUFFICIENT.
The correct answer is
A.
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