JEE 2023 — Mathematics PYQ

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Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

Let $f$ and $g$ be twice differentiable functions on $\mathbb{R}$ such that

$f''(x) = g''(x) + 6x$ ,

$f'(1) = 4g'(1) - 3 = 9$ ,

$f(2) = 3g(2) = 12$ .

Then which is NOT true?

Choose the correct answer:

Explanation

Solution:

Let $h(x) = f(x) - g(x)$ . $h''(x) = 6x \Rightarrow h'(x) = 3x^2 + C_1$ .

Given $f'(1)=9$ and $g'(1)=3 \Rightarrow h'(1) = 6 \Rightarrow 6 = 3 + C_1 \Rightarrow C_1 = 3$ .

$h'(x) = 3x^2 + 3 \Rightarrow h(x) = x^3 + 3x + C_2$ .

Given $f(2)=12, g(2)=4 \Rightarrow h(2) = 8 \Rightarrow 8 = 8 + 6 + C_2 \Rightarrow C_2 = -6$ .

$h(x) = x^3 + 3x - 6$ .

Now checking options, option (4) gives $h(-2) = -8 - 6 - 6 = -20$ , so $g(-2)-f(-2) = 20$ is true.

Option (2) $|3x^2+3| < 6 \Rightarrow 3x^2 < 3 \Rightarrow x^2 < 1 \Rightarrow -1 < x < 1$ is true. Incorrect Option: (3)