The Vitals of Mathematics

Algebra Level 5

\(a+b+c=\lambda\) for positive real variables \(a,b,c\) .
Let \(S_1=\sum \limits_{cyc} \frac{a^2}{\lambda-a}\) and \(S_2=\sum \limits_{cyc} \frac{1}{\lambda-a}\).
We define a function \(f(t)\) such that \(f(S_2)=S_1\). If \( f'(t) = 9 \) for all values of \( t \) in the domain, determine the value of \( \lambda \).

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