Show that $x$ is an eigenvector of $T$ with eigenvalue $\|T\|$

Question

Let $H$ be a Hilbert space and let $T:H\to H$ be a bounded self-adjoint linear operator. Assume there exists $x\in H$ with $\|x\|=1$ and $|\langle Tx,x\rangle|=\|T\|$.

Show that $x$ is an eigenvector of $T$ with eigenvalue $\|T\|$.

I want to show that $Tx=\|T\|x$ and I know that $\|Tx\|=|\langle Tx,x\rangle|$, but I don't know how to proceed.