If f(x) is a polynomial in x, the second derivative of f(ex) at x = 1…

If y = f(e^x) (a polynomial in e^x), then

$\frac{d y}{d x}=f'\left(e^x\right) . e^x$

$\Rightarrow \frac{d^2 y}{d x^2}=e^x f''\left(e^x\right) . e^x+e^x f'\left(e^x\right)$

$\Rightarrow \frac{d^2 y}{d x^2}=e^{2 x} f''\left(e^x\right)+e^x f'\left(e^x\right)$

$\left.\Rightarrow \frac{d^2 y}{d x^2}\right|_{x=1}=e^2 f''(e)+e f'(e)$

Hence (4) is correct answer.