JEE PYQ: Application Of Derivatives Question 7
Question 7 - 2021 (17 Mar Shift 2)
Let $f : [-1, 1] \to \mathbb{R}$ be defined as $f(x) = ax^2 + bx + c$ for all $x \in [-1, 1]$, where $a, b, c \in \mathbb{R}$ such that $f(-1) = 2$, $f’(-1) = 1$ and for $x \in (-1, 1)$ the maximum value of $f’’(x)$ is $\frac{1}{2}$. If $f(x) \le \alpha$, $x \in [-1, 1]$, then the least value of $\alpha$ is equal to ______.
Show Answer
Answer: 5
Solution
$f(-1) = a - b + c = 2$, $f’(-1) = -2a + b = 1$, $f’’(x) = 2a = \frac{1}{2}$ gives $a = \frac{1}{4}$, $b = \frac{3}{2}$, $c = \frac{13}{4}$. For $x \in [-1,1]$, $f(x) \le f(1) = \frac{1}{4} + \frac{3}{2} + \frac{13}{4} = 5$.
BETA
