Question

Show that
$\frac{1}{1 + x {}^{a - b} } + \frac{1}{1 + x {}^{b - a} } = 1$

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Answer 1

Step-by-step explanation:

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).