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JEE Main 20244 AprilEvening Shift
Exam
JEE Main
Year
2024
Date
4 April
Shift
Evening Shift
Subject
Mathematics
Chapter
Application of Derivatives

JEE Main 2024 · 4 April · Evening Shift · Mathematics · Application of Derivatives

Let f: ℝ arrow ℝ be a thrice differentiable function such that...

Let f:R→Rf: \mathbb{R} \rightarrow \mathbb{R} be a thrice differentiable function such that f(0)=0,f(1)=1,f(2)=−1,f(3)=2andf(4)=−2f(0)=0, f(1)=1, f(2)=-1, f(3)=2 and f(4)=-2. Then, the minimum number of zeros of (3f′f′′+ff′′′)(x)\left(3 f^{\prime} f^{\prime \prime}+f f^{\prime \prime \prime}\right)(x) is _______

Correct answer

0

Step-by-step solution

(3f′f′′+ff′′′)(x)=((ff′′+(f′)2)(x))′(fff′′+(f′)2)(x)=((ff′)(x))′∴(3f′f′′+f′′′)(x)=(f(x)⋅f′(x))′′\begin{aligned} & \left(3 f^{\prime} f^{\prime \prime}+f f^{\prime \prime \prime}\right)(x)=\left(\left(f f^{\prime \prime}+\left(f^{\prime}\right)^2\right)(x)\right)^{\prime} \\ & \left(f f f^{\prime \prime}+\left(f^{\prime}\right)^2\right)(x)=\left(\left(f f^{\prime}\right)(x)\right)^{\prime} \\ & \therefore\left(3 f^{\prime} f^{\prime \prime}+f^{\prime \prime \prime}\right)(x)=\left(f(x) \cdot f^{\prime}(x)\right)^{\prime \prime}\end{aligned}min⁡ roots of  f(x)→4∴min⁡. roots of f′(x)→3∴min⁡ roots of (f(x)⋅f′(x))→7∴min⁡ roots of (f(x)⋅f′(x))′′→5\begin{aligned} & \min \text{ roots\text{ of } } \mathrm{f}(\mathrm{x}) \rightarrow 4 \\ & \therefore \min \text{. roots of } \mathrm{f}^{\prime}(\mathrm{x}) \rightarrow 3 \\ & \therefore \min \text{ roots of }\left(\mathrm{f}(\mathrm{x}) \cdot \mathrm{f}^{\prime}(\mathrm{x})\right) \rightarrow 7 \\ & \therefore \min \text{ roots of }\left(\mathrm{f}(\mathrm{x}) \cdot \mathrm{f}^{\prime}(\mathrm{x})\right)^{\prime \prime} \rightarrow 5\end{aligned}$
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