#$34.$B :: Complete the following for a thin converging lens with a focal length f$.$

$($a$)$ Determine the minimum possible distance between an object and its real image. Think about what that means by drawing out the configuration; use L $=$ s $+$ s$\text{'}$ then take a derivative with respect to$$s to solve for this minimum.

$($b$)$ Once you have an equation for L in terms of s $($from above$),$ take s $=$ kf$,$ where k is a positive, real number and derive two equations: L$/$f and s$/$f$.$ Both should end as functions of k$.$ Then, graph the distance between the object and the real image as a function of the distance of the object from the lens by observing what happens to L$/$f when k changes. Use your graph to dispute or confirm the result found in part $($a$).$ Explain how and include your plot with your submission.