Given a function ' g' which has a derivative `g' (x)` for every real x and satisfies `g'(0)=2 and g(x+y)=e^y g(x)+e^y g(y)` for all x and y then:

Suppose fa n dg are functions having second derivative f'' and g' ' everywhere. If f(x)dotg(x)=1 for all xa n df^(prime)a n dg' are never zero, then (f^('')(x))/(f^(prime)(x))-(g^('')(x))/(g^(prime)(x))e q u a l (a) (-2f^(prime)(x))/f (b) (2g^(prime)(x))/(g(x)) (c) (-f^(prime)(x))/(f(x)) (d) (2f^(prime)(x))/(f(x))

Find function f(x) which is differentiable and satisfy the relation f(x+y)=f(x)+f(y)+(e^(x)-1)(e^(y)-1)AA x, y in R, and f'(0)=2.

Let f be a function with continuous second derivative and f(0)=f^(prime)(0)=0. Determine a function g by g(x)={(f(x))/x ,x!=0 0,x=0 Then which of the following statements is correct? g has a continuous first derivative g has a first derivative g is continuous but g fails to have a derivative g has a first derivative but the first derivative is not continuous

Given that f'(x) > g'(x) for all real x, and f(0)=g(0) . Then f(x) < g(x) for all x belong to (a) (0,oo) (b) (-oo,0) (c) (-oo,oo) (d) none of these

Let f be a function such that f(x+y)=f(x)+f(y) for all xa n dya n df(x)=(2x^2+3x)g(x) for all x , where g(x) is continuous and g(0)=3. Then find f^(prime)(x)dot

A twice differentiable function f(x)is defined for all real numbers and satisfies the following conditions f(0) = 2; f'(0)--5 and f"(0) = 3 . The function g(x) is defined by g(x) = e^(ax) + f (x) AA x in R , where 'a' is any constant If g'(0) + g"(0)=0 . Find the value(s) of 'a'

Suppose y=f(x) and y=g(x) are two continuous functiond whose graphs intersect at the three points (0,4),(2,2) and (4,0) with f(x) gt g(x) for 0 lt x lt 2 and f(x) lt g(x) for 2 lt x lt 4 . If int_0^4[f(x)-g(x)]dx=10 and int_2^4[g(x)-f(x)]dx=5 the area between two curves for 0 lt x lt 2 ,is (A) 5 (B) 10 (C) 15 (D) 20

For each of the function find the g _(xy), g _(yy) and g _(yx ), g (x,y) = xe ^(y)+ 3x ^(2) y

Find the partial derivatives of the functions at the indicated point G (x,y) = e ^(x+3y) log (x ^(2) + y ^(2)), (-1, 1)

For each of the function find the g _(xy), g _(yy) and g _(yx ), g (x,y) = log ( 5x + 3y)