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How many prime numbers lie in the range of the function f(x) = 7^(x⁴ + 3x² + 1)?

1. 0
2. 1
3. 2
4. Infinitely many

Correct answer: 1

Solution

The exponent g(x) = x⁴ + 3x² + 1 has minimum value 1 (at x = 0) and increases without bound. So f(x) = 7^(g(x)) has range [7, infinity). The prime numbers in this range are 7, 11, 13,... which is infinitely many. However, only 7 = 7¹ is actually achieved (at x=0); for other primes like 11, we need 7^(g(x)) = 11, but g(x) must be a real number, not necessarily an integer, so 7^(log₇(11)) = 11 is in the range. All real numbers >= 7 are in the range since f is continuous. Therefore infinitely many primes lie in [7, infinity).

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