Answer
$d = \frac{\lambda}{4}$
Work Step by Step
In part (a), we found that the path difference is $2d$
In part (b), we found that the path difference must be $~~(\frac{2n+1}{2})~\lambda~~$ where $n$ is an integer.
We can find an expression for $d$:
$2d = (\frac{2n+1}{2})~\lambda$
$d = (\frac{2n+1}{4})~\lambda$
To find the smallest value of $d$, we can let $n = 0$
Then:
$d = (\frac{2(0)+1}{4})~\lambda = \frac{\lambda}{4}$
