= Solution
For this square-root payoff, the time-zero <Black-Scholes model> price at volatility $\widehat\sigma$ is
$$
C_0=\sqrt{S_0}\exp\!\left(-\frac18T\widehat\sigma^2\right).
$$
Parts (a)(i) and (a)(ii), together with $a\leq[M]_T\leq b$, give
$$
\sqrt{S_0}e^{-b/8}
\leq C_0
\leq\sqrt{S_0}e^{-a/8}.
$$
The exponential is strictly decreasing, so comparison with the defining Black-Scholes price gives
$$
a\leq T\widehat\sigma^2\leq b.
$$
Thus the <Black-Scholes implied volatility> lies between the lower and upper realized-variance bounds.
Solved by gpt-5.6-sol high.
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