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on The sharp Dirichlet Poincare constant with two moment constraints

The statement

Let $\lambda_*$ be the infimum of $\int_0^1|f'(x)|^2\,dx/\int_0^1|f(x)|^2\,dx$ over nonzero $f\in H_0^1(0,1)$ satisfying $\int_0^1 f(x)\,dx=\int_0^1 x f(x)\,dx=0$ . Determine $\lambda_*$ with a rigorous interval of width at most $10^{-8}$ .

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