The minimum distance between an object and it's real image

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The minimum distance between an object and its real image in geometric optics, specifically using the thin lens equation, is established as 4f. To prove this mathematically, one should express the total distance d1 + d2 in terms of d1, then differentiate this equation with respect to d1. Setting the derivative equal to zero will yield the critical value of d1. After determining this value, substituting it back will provide the total distance d1 + d2. This method effectively demonstrates the relationship between object distance, image distance, and focal length.
eep
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Maybe it's just late, but I'm having an extremely difficult time proving that the minimum distance between an object and it's real image (geometric optics, thin lense equation) is 4f. I can see that it is true, however I'm unsure how to go about proving it mathematically.
 
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write an equation for the total distance d1+d2 in term of just d1 then differentiate it wrt d1 find value of d1 for which dif = zero. then find d1+d2
 

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