Let f:R R be a thrice differentiable function such that f(0)=0,f(1)=1,f(2)=-1,f(3)=2 and f…

Question

Let $f:R	o R$ 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 $(3{f}^{'}{f}^{''}+f{f}^{'''})(x)$ is \_\_\_\_\_\_\_\_\_\_.

Answer: 5

Accepted Answer

$\because f:R	o Rf(0)=0,f(1)=1,f(2)=-1$

$f(3)=2$ and $f(4)=-2$ then

$f(x)$ has atleast 4 real roots.

Then $f(x)$ has atleast 3 real roots and ${f}^{'}(x)$ has atleast 2 real roots.

Now we know that

$\begin{gathered} \frac{d}{dx}({f}^{3}\cdot {f}^{''}) =3{f}^{2}\cdot {f}^{'}\cdot {f}^{''}+{f}^{3}\cdot {f}^{'''} \ ={f}^{2}(3{f}^{'}\cdot {f}^{'}+f\cdot {f}^{''}) \end{gathered}$

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

Then its differentiation has atleast 5 distinct roots.

Solutions · 1