Let `f(x)` be a quadratic expression which is positive for all real x and `g(x)=f(x)+f'(x)+f''(x)`.A quadratic expression `f(x)` has same sign as that coefficient of `x^2` for all real`x` if and only if the roots of the corresponding equation `f(x)=0` are imaginary.Which of the following holds true? (A) `g(0)g(1)lt0` (B) `g(0)g(-1)lt0` (C) `g(0)f(1)f(2)gt0` (D) `f(0)f(1)f(2)lt0`

Let f(x) be a quadratic expression which is positive for all real x and g(x)=f(x)+f'(x)+f''(x) .A quadratic expression f(x) has same sign as that coefficient of x^2 for all real x if and only if the roots of the corresponding equation f(x)=0 are imaginary. For function f(x) and g(x) which of the following is true (A) f(x)g(x)gt0 for all real x (B) f(x)g(x)lt0 for all real x (C) f(x)g(x)=0 for some real x (D) f(x)g(x)=0 for all real x

Let f(x) be a quadratic expression which is positive for all real x and g(x)=f(x)+f'(x)+f''(x) .A quadratic expression f(x) has same sign as that coefficient of x^2 for all real x if and only if the roots of the corresponding equation f(x)=0 are imaginary.If F(x)=int_a^(x^3) g(t)dt, the F(x) is (A) an incresing function in R (B) an increasing function only in [0,oo) (C) a decreasing function R (D) a decreasing function only in [0,oo)

Let f(x) be a quadratic expression possible for all real x. If g(x)=f(x)-f'(x)+f''(x), then for any real x

If f(x) is a quadratic expression such that f(x)gt 0 AA x in R , and if g(x)=f(x)+f'(x)+f''(x) , then prove that g(x)gt 0 AA x in R .

Let g(x)= f(x) +f'(1-x) and f''(x) lt 0 ,0 le x le 1 Then

Let f(x) and g(x) be defined and differntiable for all x ge x_0 and f(x_0)=g(x_0) f(x) ge (x) for x gt x_0 then

Suppose that f(x) isa quadratic expresson 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 1 st and 2 nd derivative,respectively).g(x) 0 c.g(x)=0 d.g(x)>=0

If f(x) is a quadratic expression such that f(1) + f(2) = 0, and -1 is a root of f(x) = 0 , then the other root of f(x) = 0 is :

Let g(x)=2f((x)/(2))+f(1-x) and f'(x)<0 in 0<=x<=1 then g(x)