Maximize $\lambda$ over $a^2 + b^2 + c^2 + d^2 \ge ab + \lambda bc + cd$ and $a,b,c,d\geq 0$. Find the largest real number $\lambda$ such that
$$a^2 + b^2 + c^2 + d^2 \ge ab + \lambda bc + cd$$ for all nonnegative real numbers $a,$ $b,$ $c,$ $d.$

I tried using AM-GM on like $a^2+\dfrac{b^2}{4}\geq ab$, and I tried combining the results, but it didn't get me anywhere. Could someone give me some guidance on how to proceed?
Thanks in advance!!!

You're very much on the right track. You can use
\begin{align*}
a^2+b^2+c^2+d^2
&=\left(a^2+\frac{b^2}4\right)+\left(\frac{c^2}4+d^2\right)+\frac{3(b^2+c^2)}4\\
&\geq ab+cd+\frac{3\cdot 2bc}4
\end{align*}
to get that $\lambda=3/2$ works. Now, you just need to determine if any larger $\lambda$ will work. Can you look at the equality cases of the above process to show that any larger $\lambda$ will fail?
