Question

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

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

Hint:

When checking functional equalities with absolute values, break into sign cases for each variable and test boundary inequalities.

Answer 1

The Correct Option is C

We test the given relation:
$Ma[md(a), \ mn(a, b)]$ = maximum of $|a|$ and $\min(a, b)$.
$mn[a, \ md(Ma(a, b))]$ = minimum of $a$ and $| \max(a, b) |$.
If $a>b$, consider cases:
- If $a>0$ and $b>0$: both sides evaluate symmetrically and equality holds.
- If $a<0$ and $b<0$: both are negative, but $md(a)$ becomes positive, still symmetric.
- If $a>0$ and $b<0$: $\min(a, b) = b$ (negative), so $Ma(|a|, b) = |a|$. On the other side, $\max(a, b) = a$, so $md(\max) = |a| = a$ (since $a>0$), and $\min(a, a) = a$. This matches only if $|a| \geq |b|$.
Thus, when $a>0$, $b<0$, and $|a|<|b|$, the equality fails because LHS = $|a|$ but RHS = $a$ (smaller).
Therefore, the condition where it does not hold is $\boxed{a>0, \ b<0, \ |a|<|b|}$.