Let be a thrice differentiable function such that and . Then, the minimum number of zeros of is .

and then has atleast 4 real roots. Then has atleast 3 real roots and has atleast 2 real roots. Now we know that Here has atleast 6 roots. Then its differentiation has atleast 5 distinct roots.

Let be a thrice differentiable function such that and . Then... | JEE Main 2024 Differentiation

Q: JEE Main 2024 Differentiation: Let be a thrice differentiable function such that and . Then, the minimum number of zeros of is ....

The correct answer is 0. and then has atleast 4 real roots. Then has atleast 3 real roots and has atleast 2 real roots. Now we know that Here has atleast 6 roots. Then its differentiation has atleast 5 distinct roots.

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

$\because f: R \rightarrow R \text { and } f(0)=0, f(1)=1, f(2)=-1 \text {, }$ $f(3)=2$ and $f(4)=-2$ then $f(x)$ has atleast 4 real roots. Then $f(x)$ has atleast 3 real roots and $f^{\prime}(x)$ has atleast 2 real roots. Now we know that $\begin{aligned} \frac{d}{d x}\left(f^3 \cdot f^{\prime \prime}\right) & =3 f^2 \cdot f^{\prime} \cdot f^{\prime \prime}+f^3 \cdot f^{\prime \prime \prime} \\ & =f^2\left(3 f^{\prime} \cdot f^{\prime}+f \cdot f^{\prime \prime}\right) \end{aligned}$ Here $f^3 \cdot f'$ has atleast 6 roots. Then its differentiation has atleast 5 distinct roots.

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