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Let f(x) = sin⁶(x) + cos⁶(x). Which of the following is true?

1. f(x) belongs to [0, 1] for all real x
2. f(x) = 0 has no solution
3. f(x) belongs to [1/4, 1] for all real x
4. f(x) is an injective function

Correct answer: f(x) belongs to [1/4, 1] for all real x

Solution

Reduce the sixth powers using the sum-of-cubes identity to get a single squared sine term, then read off the range.

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