JEEOlympiad
A complex number z satisfies |z| = 1. Determine the maximum possible…
Question
A complex number z satisfies |z| = 1. Determine the maximum possible value of |z^3 - 3z + 2|. Give the answer as a decimal rounded to three places.
✓ Verified answer: 5.196checked by our engine — not a guess
Step-by-step solution
Factor: z^3 - 3z + 2 = (z - 1)^2(z + 2).
So |z^3 - 3z + 2| = |z - 1|^2 |z + 2|.
Writing z = e^{it} with c = cos t, we have |z - 1|^2 = 2 - 2c and |z + 2|^2 = 5 + 4c.
Hence f^2 = (2 - 2c)^2(5 + 4c) = 4(1 - c)^2(5 + 4c) on c in [-1, 1].
Differentiate g(c) = (1 - c)^2(5 + 4c): g'(c) = -2(1 - c)(5 + 4c) + 4(1 - c)^2 = (1 - c)(-10 - 8c + 4 - 4c) = (1 - c)(-6 - 12c).
Critical points c = 1 (gives 0) and c = -1/2.
At c = -1/2: g = (3/2)^2(3) = 27/4, so f^2 = 4*(27/4) = 27, giving f = 3*sqrt(3).
Checking the endpoint c = -1 gives f^2 = 16 < 27.
Thus the maximum is sqrt(27) = 3*sqrt(3) = 5.196.
Final answer5.196
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