TirthankarP wrote:
Which of the following expressions CANNOT have a negative value?
A) |a + b| – |a – b|
B) |a + b| – |a|
C) |2a + b| – |a + b|
D) a^2 + b^2 – 2|ab|
E) |a^3 + b^3| – a – b
Let's test each answer choice:
A) |a + b| - |a - b|
If we let a = 2 and b = -2, then |a + b| = 0, but |a - b| = 4. Thus, |a + b| - |a - b| = -4. Eliminate A.
B) |a + b| - |a|
Let a = 1 and b = -1. Then, |a + b| = 0 and |a| = 1. Thus, |a + b| - |a| = -1. Eliminate B.
C) |2a + b| - |a + b|
Let a = 1 and b = -2. Then, |2a + b| = 0, and |a + b| = 1. Thus, |2a + b| - |a + b| = -1. Eliminate C.
D) a^2 + b^2 - 2|ab|
If ab \(\geq\) 0, then |ab| = ab. In this case, the given expression becomes a^2 + b^2 - 2ab = (a - b)^2.
If ab < 0, then |ab| = -ab. In this case, the given expression becomes a^2 + b^2 - 2(-ab) = a^2 + b^2 + 2ab = (a + b)^2.
We see that the given expression is equivalent to either (a - b)^2, or (a + b)^2. Since the square of some number can never be negative, the given equation cannot be negative. This is the correct answer.
While we found our answer, let's verify that answer choice E can be negative for the sake of completeness.
E) |a^3 + b^3| - a - b
Let a = 1/2 and b = 0. Then, |a^3 + b^3| = 1/8. Substituting in the given expression, we get 1/8 - 1/2 - 0 = -3/8. We see that this expression can be negative.
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