Quadratic Equations
Quadratic Equations
star_batch_jee_advanced_2025
Grade None
Question:
Given that $a > 0$, $|ax^2 + bx + c| \leq 1$ if $-1 \leq x \leq 1$, $a, b, c \in \mathbb{R}$ and $ax + b$ has its maximum value $2$ when $-1 \leq x \leq 1$. Then $a =$
Step-by-Step Solution
Key Concept: Using function evaluations at specific points and optimization conditions uniquely determines the quadratic coefficients.
For $f(x) = ax^2 + bx + c$, use the conditions $f(0) = c$, $f(1) = a + b + c$, and $f(-1) = a - b + c$ to set up equations. From maximum value $a + b = 2$ and minimum value $f(0) = -1$, solve to get $a = 2$, $b = 0$, $c = -1$.
Correct Answer: 3