Let `g(x)=2f(x/2)+f(2-x) and f''(x) < 0 AA x in (0,2).` If g(x) increases in `(a, b)` and decreases in `(c, d),` then the value of `a + b+c+d-2/3` is

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 f(x) = x^(2) + 2x +5 and g(x) = x^(3) - 1 be two real functions. Find (f+g)(x), (f-g)(x), (fg)(x) and ((f)/(g))(x) .

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 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) be twice differentiablesuchthat f''(x)=-f(x),f'(x)=g(x), where f'(x) and f''(x) represent the first and second derivatives of f(x), respectively.Also if h(x)=(f(x))^(2)+(g(x))^(2) and h(5)=5 then h(10) is equal to

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