Let `g(x)=2f(x/2)+f(2-x)` and `f''(x)<0`,`AA x in``(0,2)` .Then `g(x)` increasing in

Let g(x) = 2 f(x/2)+ f(2-x) and f''(x) lt 0 forall x in (0,2) . Then calculate the interval in which g(x) is increasing.

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

Let f(x+y)=f(x) f(y) and f(x)=1+(sin 2x)g(x) where g(x) is continuous. Then, f'(x) equals

Let g(x)=(f(x))^(2)+f(x)+19 where f(x)=|x-2|^(2)+|x-2|-2 then number of single digit natural number of the range of g(x) is

Let f(x+y)=f(x)+f(y) and f(x)=x^(2)g(x)AA x,y in R where g(x) is continuous then f'(x) is

Let f_(1) (x) and f_(2) (x) be twice differentiable functions where F(x)= f_(1) (x) + f_(2) (x) and G(x) = f_(1)(x) - f_(2)(x), AA x in R, f_(1) (0) = 2 and f_(2) (0) = 1. "If" f'_(1)(x) = f_(2) (x) and f'_(2) (x) = f_(1) (x) , AA x in R . then the number of solutions of the equation (F(x))^(2) =(9x^(4))/(G(x)) is...... .

Let f (x) and g (x) be two differentiable functions, defined as: f (x)=x ^(2) +xg'(1)+g'' (2) and g (x)= f (1) x^(2) +x f' (x)+ f''(x). The value of f (1) +g (-1) is: