Multiple choice

Let f(x) be a quadratic polynomial with a positive leading coefficient such that f(x) is non-negative for all real numbers x. Given f(1) = 0 and f(4) = 8, determine the value of 9f(9).

  1. 427

  2. 436

  3. 512

  4. 556

  5. 565
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C Correct answer
Explanation

f(x) is a quadratic with positive leading coefficient and f(x) >= 0. Since f(1)=0, x=1 is a root. Since f(x)>=0, the vertex must be at the root, so f(x) = a(x-1)^2. Given f(4)=8, a(4-1)^2 = 8 => 9a = 8 => a = 8/9. So f(x) = (8/9)(x-1)^2. Then 9f(9) = 9 * (8/9)(9-1)^2 = 8 * 8^2 = 8 * 64 = 512.

AI explanation

Since f(x) has a positive leading coefficient and is non-negative for all real numbers, f(1) = 0 must be the vertex of the parabola, meaning its equation is f(x) = a(x - 1)^2. Using f(4) = 8 gives 8 = a(4 - 1)^2, so 9a = 8 and a = 8/9. For x = 9, f(9) = 8/9 * (9 - 1)^2 = 8/9 * 64 = 512/9. Therefore, 9f(9) = 9 * 512/9 = 512.

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