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Is A positive? 1. x^2-2*x+A is positive for all x 2. A*x^2+1 is positive for all x

The questioin is not clear !!!

1) is insufficient since x(x-2)>-A With different x A changes

if we say CASE1 1. x^2-2*x+A is positive for all x INSUFFI 2. A*(x^2)+1 is positive for all x SUFFI different result !!! else if we say CASE 2 1. x^2-2*x+A is positive for all x INSUFFI 2. (A*x)^2+1 is positive for all x INSUFFI different result !!!

in second case CASE2 above ,(2) is INSUFFI since A can be +ve or -ve

In case1 ,(2) is SUFFI ,A needs to be +ve -ve value cannot suffice !! _________________

1. x^2-2*x+A is positive for all x 2. A*x^2+1 is positive for all x

Remember for all x!!!

In statement 1, can anybody prove that A is -ve if 0 < x < 2?

Therefore A.

Hi, why are you limiting to 0<x<2?

My solution. 1) simplifies to A>x(2-x). A is +ve for any value 0<x<2; and could be +ve or -ve for all other values. Insufficient. 2) simplifies to A > -1/(x^2); A can be +ve or -ve for any value of x (except for x=0); therefore Insufficient. 1 and 2 together also insufficient, e.g. at x=3, (1) gives A>-3 and (2) gives A>-1/9; therefore A could still be either -ve or +ve; Answer E

PS I'm not sure if this helps visualize, as I don't know how to explain it ... I see (1) as an parabola cutting x axis at 0 and 2, center at 1,1; and (2) as a kind of hyperbola below the x axis.

Every one who are sticking with E is missing my point. Try to understand what exactly meant by "for all x ". You never agree with me on OA (as A) if you keep on missing this statement: "for all x ".

If A is +ve, that value works for all x. If A is -ve, that value doesnot work for all x.

Every one who are sticking with E is missing my point. Try to understand what exactly meant by "for all x ". You never agree with me on OA (as A) if you keep on missing this statement: "for all x ".

If A is +ve, that value works for all x. If A is -ve, that value doesnot work for all x.

Every one who are sticking with E is missing my point. Try to understand what exactly meant by "for all x ". You never agree with me on OA (as A) if you keep on missing this statement: "for all x ".

If A is +ve, that value works for all x. If A is -ve, that value doesnot work for all x.

Therefore A must be +ve.

If any, will clearify again.

from statement 1:

(x^2 - 2x + A) > 0 (x^2 - 2x +1) + (A -1) > 0 (x-1)^2 + (A -1) > 0

suppose if x = 1, the inequality becomes: (A -1) > 0. so A has to be >1 to satisfy the inequality. Therefore A must be +ve.

The easiest way to understand this problem is that "when 1 > x > -1, A must be positive; when x>1 but <-1, A can either be positive or negative. Therefore the value of A has to be +ve for all values of x. Hence A is positive