Tightness of the descent lemma

The descent lemma states that for $L$-smooth $f$,

$$f(y) \le f(x) + \langle \nabla f(x), y-x\rangle + \frac L2\|y-x\|_2^2 .$$

Exhibit a function for which this holds with equality for all $x,y$, and deduce that the constant $L/2$ cannot be improved. Then show that the step size $\eta = 1/L$ is exactly the minimiser of the right-hand side over steps of the form $y = x - \eta\nabla f(x)$.