1. For a string fixed at both ends, stationary waves are formed only when the length of the string contains an integer number of half-wavelengths:

$L = n\frac{\lambda}{2}, \qquad n=1,2,3,\dots$

2. Here, the string length is

$L = 40\text{ cm}$

3. The maximum wavelength corresponds to the fundamental mode, i.e. the smallest possible value of $n$:

$n=1$

So,

$40 = \frac{\lambda_{\max}}{2}$

$\lambda_{\max} = 2\times 40 = 80\text{ cm}$

4. Therefore, the maximum wavelength of the stationary wave is

$\boxed{80\text{ cm}}$

5. Checking options:

• A: $20$ ❌
• B: $80$ ✅
• C: $40$ ❌
• D: $120$ ❌

Hence, the correct option is B.