Question

Let $y=f\left(x\right)$ be a thrice differentiable function defined on R such that f(x) = 0 has atleast 7 distinct zeros, then minimum number of zeros of the equation $f\left(x\right)+9f^{{}^{\prime }}\left(x\right)+27f^{{}^{\prime \prime }}\left(x\right)+27f^{{}^{\prime \prime \prime }}\left(x\right)=0$ is

Answer 1

Let $g\left(x\right)=e^{\frac{x}{3}}f\left(x\right)$
then $g\left(x\right)=0$ has at least 7 distinct zero and using rolle's theorem $g^{{}^{\prime \prime \prime }}\left(x\right)=0$ has at least 4 distinct zeroes.
$g^{{}^{\prime }}\left(x\right)=e^{\frac{x}{3}}{f}^{{}^{\prime }}\left(x\right)+\frac{e^{\frac{x}{3}}}{3}f\left(x\right)$
$g^{{}^{\prime \prime }}\left(x\right)=\frac{e^{\frac{x}{3}}}{9}(6f^{\prime }+f+gf")$
$g^{{}^{\prime \prime \prime }}\left(x\right)=\frac{e^{\frac{x}{3}}}{27}\left(f\right(x)+9f^{{}^{\prime }}(x)+27f^{\prime \prime }(x)+27f^{{}^{\prime \prime \prime }}(x\left)\right)$