How large can mutual inductance be?

Two fixed linear coils have self-inductances 9.0 mH and 16.0 mH. Their coupling is passive, and the magnetic energy must be nonnegative for every pair of currents.

What is the largest possible magnitude of their mutual inductance?

Organize the givens & assumptions

L₁ = 9.0 mH
L₂ = 16.0 mH
Nonnegative energy for every current pair

Linear reciprocal magnetic system without an active element or changing geometry. The sign of M depends on reference directions; the question asks its magnitude.

A $25\ \mathrm{mH}$
B $7\ \mathrm{mH}$
C $12\ \mathrm{mH}$
D $144\ \mathrm{mH}$

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