Suppose f(x) is such a quadrant expression that it is positive for all real x . If g(x)=f(x)+f^ (x)+f^ (x) , then for anyreal x . Then for any real x .

Step-by-step solution

Let f(x)=a x^2+b x+c, a 0, b^2 4 a c ( f(x) 0) Now, g(x)=a x^2+b x+c+2 a x+b+2 a=a x^2+(b+2 a) x+2 a+b+c Now, (b+2 a)^2-4 a(2 a + b + c ) aligned & =b^2+4 a b+4 a^2-8 a^2-4 a b-4 a c & =b^2-4 a c-8 a^2 0 & g(x) 0 aligned