Question

\( \text{Given that } a > b \text{ then the relation } Ma[md(a), mn(a,b)] = mn[a, md(Ma(a,b))] \text{ does not hold if} \)

• A. \(a > 0,\ b < 0,\ |a| < |b|\)
• B. \(a > 0,\ b > 0\)
• C. \(a < 0,\ b < 0\)
• D. \(a > 0,\ b < 0,\ |a| > |b|\)

Hint:

Since a > b, Ma(a, b) = a and mn(a, b) = b. Simplify both sides, then test each sign case.

Answer 1

The Correct Option is C

Step 1: Simplify both sides using a > b:
Ma(a, b) = a and mn(a, b) = b. Left side = Ma[|a|, b] = the larger of |a| and b. Right side = mn[a, |a|] = the smaller of a and |a|.

Step 2: Case a < 0, b < 0 (with a > b):
|a| = -a > 0 and b < 0, so the left side = |a| = -a, a positive number. The right side = smaller of a and |a| = a, a negative number. They are different, so the relation does not hold.

Step 3: Case a > 0, b > 0:
|a| = a. Left side = larger of a and b = a (as a > b). Right side = smaller of a and a = a. The relation holds.

Step 4: Cases a > 0, b < 0:
|a| = a, and b is negative, so the left side = larger of a and b = a. The right side = smaller of a and a = a. The relation holds whether |a| < |b| or |a| > |b|, so options (1) and (4) do not break it.

Final Answer:
The correct option is (3): \(a < 0,\ b < 0\)