QJEEM24Mac495: JEE Main 2024 Mathematics - Class 12 JEEM24Mac495 Solution: Let f : R → R be a thrice differentiable function such that f ( …

$$\because f:R\to R\text{ and }f(0)=0,f(1)=1,f(2)=-1\text{, }$$

$$f(3)=2$$ 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

$$\begin{array}{rl}\frac{d}{dx}(f^3\cdot f^{\prime \prime })& =3f^2\cdot f^{\prime }\cdot f^{\prime \prime }+f^3\cdot f^{\prime \prime \prime }\\ & =f^2(3f^{\prime }\cdot f^{\prime }+f\cdot f^{\prime \prime })\end{array}$$

Here $$f^3\cdot f^{\prime }$$ has atleast 6 roots.

Then its differentiation has atleast 5 distinct roots.