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Show that the equation (arcsin x)³ + (arccos x)³ = a*pi³ has no solution when a < 1/32 or a > 7/8.

1. Proof: range of LHS / a is [1/32, 7/8], so no roots outside this interval
2. Proof: range of LHS / a is [1/32, 1/8]
3. Proof: range of LHS / a is [1/16, 7/8]
4. Proof: range of LHS / a is [0, 1]

Correct answer: Proof: range of LHS / a is [1/32, 7/8], so no roots outside this interval

Solution

Set s = arcsin x in [-pi/2, pi/2]; then arccos x = pi/2 - s. The expression becomes a function of s whose extreme values determine the attainable range of a. Outside [1/32, 7/8] there is no solution.

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