The function $f(x) = ax^2 - c$ satisfies $-4 \leq f(1) \leq -1$ and $-1 \leq f(2) \leq 5$. Which of the following statements is true?
Step-by-Step Solution
Key Concept: The quadratic structure allows expressing function values at any point as linear combinations of values at other points.
By using derivative relations $a = \frac{1}{2}[f(2) - f(1)]$, $c = \frac{1}{3}[f(2) - 4f(1)]$, derive that $f(3) = 9a - c = \frac{1}{8}[f(2) - 5f(1)]$, expressing the quadratic's value at a new point via values at previously evaluated points.
Correct Answer: 1,2,3,4