For what values of $\rho$ is $\lambda \geq 0$ in $(1-\lambda)(\lambda^2 - 2\lambda + 1 - \rho) - 0.5( 0.5 - 0.5\lambda - 0.5\rho)$?

Question

Consider the expression

$$ (1-\lambda)(\lambda^2 - 2\lambda + 1 - \rho) - 0.5( 0.5 - 0.5\lambda - 0.5\rho) = 0 $$

We seek the entire range of values of $\rho$ such that $\lambda \geq 0$ in the above expression. Note that the constraints on $\rho$ is $-1 \leq \rho \leq 1$.

I plugged in the lower bound for $\lambda$, i.e., $\lambda = 0$ and obtained $\rho = 1$. So this gives us an upper bound. How do we get a lower bound?

Answer 1

Hint: Set $\mu=1-\lambda$.

In this way, your expression becomes:

$$\mu(\mu^2-\rho)-\tfrac14(\mu-\rho)=0 \tag{1}$$

with the new question: with constraint (1) for which value of $\rho \in (-1,1)$ do we have $\mu<1$ ?

(1) can be written:

$$\rho=\underbrace{\dfrac{\mu(\mu^2-\tfrac14)}{\mu-\tfrac14}}_{f(\mu)}\tag{2}$$

Now, conclude from the graphical representation of $f$: