Min and Max values of Bayesian Risk Classifier when Posteriori Probability and PDFs are Unknown

I'm struggling to come up with a well reasoned argument for this problem.

Let $$\tau_1$$ be the posteriori probability and let $$L(r^*)$$ be the risk classifier.

For this scenario, assume:

$$X\in\mathbb{X}=[0,1],Y\in\{ 0,1 \}$$

$$\pi_y=P(Y=y)=1/2$$ for $$y\in{0,1}$$

Now assume $$P(Y=1|x)=\tau_1\in[0,1]$$ is unknown and the PDFs $$f(x|Y=0)$$ and $$f(x|Y=1)$$ are also unknown. In other words the only things that are known is that $$f(\centerdot|Y=y):\mathbb{X}\longrightarrow\mathbb{R}$$ is a density function on $$\mathbb{X}=[0,1]$$ for each $$y\in\{0,1\}$$. In other words $$f(x|Y=y)\ge0$$ for all $$x\in\mathbb{X}$$ and $$\int_\mathbb{X}f(x|Y=y)dx=1$$.

From this information I'd like to deduce the min and max values of $$L(r^*)$$ and provide conditions on $$\tau_1$$, $$f(\centerdot|Y=0)$$ and $$f(\centerdot|Y=1)$$ in order to yield those values.

• Take my answer for the first part here: stats.stackexchange.com/a/549963/144600 and find the extrema points by derivation and equating to 0. Oct 27 '21 at 8:22
• @Spätzle: Please note that we want the OP to add self-study tag themselves, see stats.meta.stackexchange.com/questions/5611/… Better to ask a standard comment as I do below Oct 27 '21 at 11:44
• Please add the self-study tag & read its wiki. Then tell us what you understand thus far, what you've tried & where you're stuck. We'll provide hints to help you get unstuck. Please make these changes as just posting your homework & hoping someone will do it for you is grounds for closing. Oct 27 '21 at 11:45
• @kjetilbhalvorsen this is one question in a series of many by this OP, all with the same setting and an increasing level of abstraction. He/she have made it clear its a class material they didn't manage to understand. Oct 27 '21 at 13:23