If `f"(x)>0AAx in R, f'(3)=0,and g(x)=f("tan"hat2x-2"tan"x+4),0

If f''(x) gt forall in R, f(3)=0 and g(x) =f(tan^(2)x-2tanx+4y)0ltxlt(pi)/(2) ,then g(x) is increasing in

If f"(x) > 0 and f(1) = 0 such that g(x) = f(cot^2x + 2cotx + 2) where 0 < x < pi , then g'(x) decreasing in (a, b). where a + b + pi/4 …

If f(0) = f'(0) = 0 and f''(x) = tan^(2)x then f(x) is

For the function f(x)=(x^2+b x+c)3^xa n dg(x)=(x^2+b x+c)3^x+3^x(2x+b)dot Which of the following holds goods? If f(x)>0AAx in R =>g(x)>0AAx in R If f(x)>0AAx in R=>g(x) 0AAx in R=>f(x)>0AAx in R If g(x)>0AAx in R =>f(x)<0AAx in R

Let f and g be differentiable functions such that: xg(f(x))f\'(g(x))g\'(x)=f(g(x))g\'(f(x))f\'(x) AA x in R Also, f(x)gt0 and g(x)gt0 AA x in R int_0^xf(g(t))dt=1-e^(-2x)/2, AA x in R and g(f(0))=1, h(x)=g(f(x))/f(g(x)) AA x in R Now answer the question: f(g(0))+g(f(0))= (A) 1 (B) 2 (C) 3 (D) 4

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 a real valued function f satisfy f(x + y) = f(x)f(y)AA x, y in R and f(0)!=0 Then g(x)=f(x)/(1+[f(x)]^2) is

Let f'(sinx)lt0andf''(sinx)gt0,AAx in (0,(pi)/(2)) and g(x)=f(sinx)+f(cosx), then find the interval in which g(x) is increasing and decreasing.

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...... .

f:[0,5]rarrR,y=f(x) such that f''(x)=f''(5-x)AAx in [0,5] f'(0)=1 and f'(5)=7 , then the value of int_(1)^(4)f'(x)dx is