If f(2x)=4x^{2}+1, then for how many <strong>REAL</strong> values of x will f(2x) be the GM of f(x) and f(4x)?
- A. Four
- B. Two
- C. One ✓
- D. None
Correct Answer: C. One
Explanation
Substituting x for 2x gives f(x) = x^2 + 1. For f(2x) to be the geometric mean, (f(2x))^2 = f(x)f(4x), leading to (4x^2+1)^2 = (x^2+1)(16x^2+1). Expanding gives 16x^4 + 8x^2 + 1 = 16x^4 + 17x^2 + 1. This simplifies to 9x^2 = 0, giving x = 0. Thus, there is exactly one valid real value.
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