Finding maximum value of $|f(z)|$ using Maximum modulus theorem? In the question asked here→ Maximum Modulus Exercise

I want to know, if we just want to find maximum value of $|f(z)|$, why 'Marlu' Sir in his answer (here https://math.stackexchange.com/a/325832/168676) has done calculations to show that there are no other Maxima?

My attempt: as $f(z)=z^2-3z+2$ is analytic inside and on $|z|=1$ hence by maximum modulus theorem, maximum value of $|f(z)|$ occurs on boundary! 
and by traingle inequality, 

$|f(z)|=|z^2-3z+2|≤|z^2|+3|z|+2$

$$≤6$$ (since on the boundary, $|z|=1$)

So from here, we know, maximum value of $|f(z)|$ cannot exceed $6$ and the fact that,  at point  $z=-1$ which is on boundary, $f(-1)=6$ confirms that, maximum value of $|f(z)|$ is $6$ is am I correct? Please help me...
No, 6 is just an upper bound but not the maximum modulus, check out the zill-complex analysis book.

Check out my annotated image if you still have issue.
No, 6 is just an upper bound but not the maximum modulus, check out the zill-complex analysis book.

Check out my annotated image if you still have issue.