Suppose that f (x) is a quadratic expression positive for all real x. If g(x)=f(x)+f′(x)+f′′(x), then for any real x (where f'(x) and f"(x) represent 1st and 2nd derivative, respectively)

Let f(x)=ax2+bx+c be a quadratic expression such thatf(x)>0 for all x∈R. Then, a>0 and b2−4ac<0. Now, g(x)=f(x)+f′(x)+f′′(x)⇒ g(x)=ax2+x(b+2a)+(b+2a+c)Discriminant of g(x) isD=(b+2a)2−4a(b+2a+c)=b2−4a2−4ac=b2−4ac−4a2 <0∵b2−4ac<0Therefore, g(x) > 0 for all x∈R.