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What if in statement 2: (1) x = 0.5, A = 1 and (2) x = 0.5, A = -1?

It is "for all x". Hence, A has to be positive for all x. For x = 0.5, A can be negative , but for x = 50, it cannot be. For all values of x, if A is positive, the inequality will hold good.

What if in statement 2: (1) x = 0.5, A = 1 and (2) x = 0.5, A = -1?

It is "for all x". Hence, A has to be positive for all x. For x = 0.5, A can be negative , but for x = 50, it cannot be. For all values of x, if A is positive, the inequality will hold good.

exactly. here A doesnot need to be +ve to have (Ax^2 + 1) > 0.

With -ve and +ve values for A, (Ax^2 + 1) has to be +ve but not necessarily A.
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exactly. here A doesnot need to be +ve to have (Ax^2 + 1) > 0.

With -ve and +ve values for A, (Ax^2 + 1) has to be +ve but not necessarily A.

I will begin with stmt1.

From stmt1: (x-1)^2 + (A-1) > 0 Here, (x-1)^2 is positive for all X. And, A can be positive or negative for the expression to be true.

For example, x = 0.5, A = 1. or, x = 5, A = -10 Both the cases will satisfy the ineuqality. However, for certain x, A can be +ve or -ve, but for some other x, A has to be +ve. That means, A must be positive for all x.

From stmt2: Ax^2 + 1 > 0 Here, second expression is positive and x^2 is positive. Hence, A can be positive or negative for certain x and only positive for other x. For example, for x = 0.5, A can be -1 or +1. but, for x = 5, A has to be +1 only. But, positive value of A will satisfy all values of x and not the negative value.

exactly. here A doesnot need to be +ve to have (Ax^2 + 1) > 0.

With -ve and +ve values for A, (Ax^2 + 1) has to be +ve but not necessarily A.

I will begin with stmt1.

From stmt1: (x-1)^2 + (A-1) > 0 Here, (x-1)^2 is positive for all X. And, A can be positive or negative for the expression to be true.

For example, x = 0.5, A = 1. or, x = 5, A = -10 Both the cases will satisfy the ineuqality. However, for certain x, A can be +ve or -ve, but for some other x, A has to be +ve. That means, A must be positive for all x.

From stmt2: Ax^2 + 1 > 0 Here, second expression is positive and x^2 is positive. Hence, A can be positive or negative for certain x and only positive for other x. For example, for x = 0.5, A can be -1 or +1. but, for x = 5, A has to be +1 only. But, positive value of A will satisfy all values of x and not the negative value.

I thik it would be better if we say A can be +ve or 0.

I agree with you that A cannot be -ve.
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