Most important concept here is that man moves slowly. Slowly means, always in equilibrium
For the man,
N + T sinθ - Mg = 0 (vertical)
F -T cosθ = 0 (horizontal)
Maximum extension is obtained when static friction on man is maximum
For maximum extension, f = µN
For spring, T = cosθ - kx = 0 (horizontal)
Solve for T : T = cosθ kx ⇒ T = kx / cosθ
Substitute for F and solve for N
µN - T cosθ = 0
OrN = Tcosθ / µ - kx/µ
+
= Mg
kx (1 + µtanθ) = µMg
x =