Quadratic Equations
Quadratic Equations
star_batch_jee_advanced_2025
Grade 11
Question:
Let $f(x) = ax^2 + bx + c$, $a, b, c \in \mathbb{R}$. Suppose $|f(x)| \leq 1$, $\forall x \in [0,1]$ then:
$|a| \leq 8$
$|a + 2b + 4c| \leq 4$
$|a| + |b| + |c| \leq 17$
$|3a + 2b| \leq 8$
Step-by-Step Solution
Key Concept: Use specific test points on a quadratic to construct linear inequalities on coefficients, then combine them systematically.
By substituting $x=0,1,\frac{1}{2}$ into $f(x)$ where $|c|\leq 1$, $|a+b+c|\leq 1$, and $|\frac{a}{4}+\frac{b}{2}+c|\leq 1$, we derive the system of inequalities $-1\leq c\leq 1$, $-1\leq a+b+c\leq 1$, and $-4\leq a+2b+4c\leq 4$. Rearranging yields $|a|\leq 8$, $|b|\leq 8$, $|c|\leq 1$.
Correct Answer: 1,2,3,4