Inequality regarding matrix norm and positive definite matrix

Question

Is the following inequality true $$\frac{x^TB^TABx}{x^TAx}\le \Vert B \Vert^2$$ where A is a positive definite matrix, B is an arbitrary matrix, x is a column vector, and $\Vert\cdot\Vert$ is the matrix 2-norm. It seems true to me, and I've verified a lot of cases numerically, but I don't know how to prove it.

Answer 1

Try $$A=\begin{bmatrix}10&0\\0&1\end{bmatrix},\ \ \ B=\begin{bmatrix}0&1\\1&0\end{bmatrix},\ \ \ x=\begin{bmatrix}0\\1\end{bmatrix}. $$

Answer 2

The answer is that this is not necessarily true. For instance, take $$ A = \pmatrix{1&0\\0&4}, \quad B = \pmatrix{0&1\\1&0}, \quad x = \pmatrix{1\\0}. $$