Let $$f\left( x \right)$$  be a polynomial function of degree 2 and $$f\left( x \right) > 0$$   for all $$x\, \in \,R.$$  If $$g\left( x \right) = f\left( x \right) + f'\left( x \right) + f''\left( x \right)$$       then for any $$x\,:$$

Solution :
$$\eqalign{ & {\text{Let }}f\left( x \right) = a{x^2} + bx + c \cr & {\text{Then }}g\left( x \right) = a{x^2} + bx + c + 2ax + b + 2a \cr & = a{x^2} + \left( {2a + b} \right)x + 2a + b + c \cr & {\text{As }}f\left( x \right) > 0\,{\text{for all }}x, \cr & a > 0\,{\text{and }}D = {b^2} - 4ac < 0 \cr & g\left( x \right) > 0\,{\text{for all }}x, \cr & {\text{if }}a > 0\,{\text{and }}D = {\left( {2a + b} \right)^2} - 4a\left( {2a + b + c} \right) < 0 \cr & {\text{Now, }}{\left( {2a + b} \right)^2} - 4a\left( {2a + b + c} \right) = - 4{a^2} + {b^2} - 4ac < 0{\text{ }} \cr & {\text{Hence, }}g\left( x \right) > 0 \cr} $$