 Solution

Fermat's principle makes the optical path length, or transit time in a stationary medium, stationary under small variations of the light ray with fixed endpoints. It need not be a global minimum. In the result of part (a), the coefficient of $d^2$ isFor a family of paraxial light rays from the object to its image, stationarity with respect to $d$ requires this coefficient to vanish. Therefore the thin lens equation isFor a distant object, $a\to \infty$, and the image position $b=f$ defines the focal length. HenceThe axial member $d=0$ is stationary by symmetry for arbitrary $b$; requiring the neighbouring non-axial light rays to meet the same image is what determines $b$.