Let $y=f\left(x\right)$ be a continuous and twice differentiable function such that $f\left(1\right)=f\left(3\right)=f\left(5\right)=0$, then which is/are CORRECT ?

(a) Consider $\mathrm{g}\left(\mathrm{x}\right)=\mathrm{f}\left(\mathrm{x}\right){\mathrm{e}}^{-\left(2\mathrm{x}\right)}$

since g(1) = g(3) = g(5) = 0

$\Rightarrow \mathrm{g}\text{'}\left(\mathrm{x}\right)=0$ has atleast two roots in (1, 5)

$\Rightarrow {\mathrm{e}}^{-2\mathrm{x}}(\mathrm{f}\text{'}\left(\mathrm{x}\right)-2\mathrm{f}\left(\mathrm{x}\right))=0$

$\Rightarrow \mathrm{f}\text{'}\left(\mathrm{x}\right)=2\mathrm{f}\left(\mathrm{x}\right)$ , has atleast two roots

(c) Consider $\mathrm{g}\left(\mathrm{x}\right)=\mathrm{f}\left(\mathrm{x}\right){\mathrm{e}}^{-\left(2\mathrm{x}\right)}$

since, g’(x) = 0, has a real root

$\Rightarrow \mathrm{g}"\left(\mathrm{x}\right)=0$has a real root

$\Rightarrow \mathrm{f}"\left(\mathrm{x}\right)-4\mathrm{f}\text{'}\left(\mathrm{x}\right)+4\mathrm{f}\left(\mathrm{x}\right)=0$, has a real root

(d) Apply Rolle’s theorem for f(x) in [1, 3] & [ 3, 5]