1. The problem statement, all variables and given/known data Find the maximum value of |(z-1)(z+1/2)| for |z|≤1. 2. Relevant equations Calculus min/max concepts? 3. The attempt at a solution Let f(z)=|(z-1)(z+1/2)|. Observe f(z) is the product of 2 analytic functions on |z|≤1, g(z)=z-1 and h(z)=z+1/2. Therefore f(z) is analytic on |z|≤1. Since f(z) is analytic on |z|≤1, it is continuous on the same domain. f'(z)=2z-1/2. Critical points at z=1 and 1/4. f(1)=0 f(1/4)=9/16 The maximum value of |(z-1)(z+1/2)| for |z|≤1 is 9/16. Is this close? I suspect I'm missing something fundamental as this doesn't use any complex analysis content.