Suppose that `f(x)` is a quadratic expresson positive for all real `xdot` If `g(x)=f(x)+f'(x)+f''(x),` then for any real `x(w h e r ef'(x)a n df''(x)` represent 1st and 2nd derivative, respectively). a. `g(x)<0` b. `g(x)>0` c. `g(x)=0` d. `g(x)geq0`

Suppose that f(x) isa quadratic expresson positive for all real xdot If g(x)=f(x)+f^(prime)(x)+f^(prime prime)(x), then for any real x

Suppose that f(x) is a quadratic expresson positive for all real xdot If g(x)=f(x)+f^(prime)(x)+f^(primeprime)(x), then for any real x (where f^(prime)(x) and f^(primeprime)(x) represent 1st and 2nd derivative, respectively). (a) g(x) 0 (c). g(x)=0 (d). g(x)geq0

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 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

If f(x)=e^(x)g(x),g(0)=2,g'(0)=1, then f'(0) is

f'(x) = g(x) and g'(x) =-f(x) for all real x and f(5)=2=f'(5) then f^2 (10) + g^2 (10) is

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

Let g(x)=1+x-[x]a n df(x)={-1, x 0 . Then for all x ,f(g(x)) is equal to (where [.] represents the greatest integer function). (a) x (b) 1 (c) f(x) (d) g(x)

Let f(x)a n dg(x) be differentiable functions such that f^(prime)(x)g(x)!=f(x)g^(prime)(x) for any real xdot Show that between any two real solution of f(x)=0, there is at least one real solution of g(x)=0.

If f(x+y)=f(x)dotf(y) for all real x , ya n df(0)!=0, then prove that the function g(x)=(f(x))/(1+{f(x)}^2) is an even function.