The minimum distance between an object and its real image formed by a convex lens of focal length f is

Step-by-step solution

We know: Let the seperation between object and image by x , x=u-v Using equation (1), on rewriting, aligned & x=u- x f u & x= u^2 (u+f) ---(1) aligned we first take log of equation (1) (x)=2 (u)- (u+f) Subsequently, to minimize x w.r.t. u , take the derivative of x w.r.t. u : aligned & 1 x d x d u = 2 u - 1 (u+f) & d x d u = x(u+2 f) u(u+f) ---(2) aligned Condition for minimum x requires, d x d u =0 and d^2 x u^2 <0 d x d u = x(u+2 f) u(u+f) =0 So, x=0 is the trivial solution and the real solution occurs for u=-2 f