# 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 \( \lambda \).