Greatest Value Of Modulus Sum

Question

Among all complex numbers z lying on the circle defined by |z| = 1, the maximum value attained by the expression |z^2 - z + 1| over this unit circle equals which number?

RankGuru · Accepted Answer

Maximizing the modulus of a polynomial in z on the unit circle is a JEE Advanced extremal problem best handled by writing z = e^{iθ} and reducing to a real trigonometric function. On |z| = 1 we have z·z-bar = 1, so z-bar = 1/z. Consider |z^2 - z + 1|. Factor out z: z^2 - z + 1 = z(z + 1/z - 1) = z((z + z-bar) - 1) since 1/z = z-bar on the unit circle. With z + z-bar = 2cosθ, the bracket becomes (2cosθ - 1), a real number, and |z| = 1, so |z^2 - z + 1| = |2cosθ - 1|. As θ varies, 2cosθ ranges over [-2, 2], so 2cosθ - 1 ranges over [-3, 1], whose maximum absolute value is 3, achieved at cosθ = -1, that is z = -1. Option 1 takes only the upper end of the bracket. Option 2 ignores the constant shift. Option 13/4 confuses this with a different extremum. Hence the maximum is 3. Plausibility check: at z = -1, z^2 - z + 1 = 1 + 1 + 1 = 3, directly attaining the bound and confirming the trigonometric reduction.

Greatest Value Of Modulus Sum

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