Question

if a,b,c are all positive , show that all roots of 1/x-a+1/x-b+1/x-c=1/X are real​

Answer 1

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Step-by-step explanation:

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Here's what some of the graphs look like. This is not a proof but, perhaps, it will give you some clue as to what's going on, particularly if you can show that the generic graph looks like this. Thus, you might think of this as the geometric motivation behind some of the other answers that encourage you to think about where the function behaves in some fashion

Let f(x)=(x−a)(x−b)(x−c). Then

f′(x)f(x)=1x−a+1x−b+1x−c.

Thus, the problem asks you to prove that f′(x)=0 has exactly two real roots not equal to a,b,c.

Since f(x) is cubic, f′ is quadratic, thus it has at most two real rules.

By Rolle Theorem, f′ has a root in (a,b) and a root in (b,c).