Suppose that `f(0)=0a n df^(prime)(0)=2,` and let `g(x)=f(-x+f(f(x)))dot` The value of `g'` (0) is equal to _____

If for a continuous function f,f(0)=f(1)=0,f^(prime)(1)=2a n dy(x)=f(e^x)e^(f(x)) , then y^(prime)(0) is equal to a. 1 b. 2 c. 0 d. none of these

Let f(x) be differentiable for real x such that f^(prime)(x)>0on(-oo,-4), f^(prime)(x) 0on(6,oo), If g(x)=f(10-2x), then the value of g^(prime)(2) is a. 1 b. 2 c. 0 d. 4

Let g^(prime)(x)>0a n df^(prime)(x) g(f(x-1)) f(g(x+1))>f(g(x-1)) g(f(x+1))

Let fa n dg be differentiable on R and suppose f(0)=g(0)a n df^(prime)(x)lt=g^(prime)(x) for all xgeq0. Then show that f(x)lt=g(x) for all xgeq0.

Let x = f(t) and y = g(t), where x and y are twice differentiable function If f^(')(0) = g^(')(0) = f^('')(0) = 2, g^(")(0) = 6 , then the value of ((d^(2)y)/(dx^(2))_(t=0) is equal to

Let f(x0 be a non-constant thrice differentiable function defined on (-oo,oo) such that f(x)=f(6-x)a n df^(prime)(0)=0=f^(prime)(x)^2=f(5)dot If n is the minimum number of roots of (f^(prime)(x)^2+f^(prime)(x)f^(x)=0 in the interval [0,6], then the value of n/2 is___

Suppose that f(x) is differentiable invertible function f^(prime)(x)!=0a n dh^(prime)(x)=f(x)dot Given that f(1)=f^(prime)(1)=1,h(1)=0 and g(x) is inverse of f(x) . Let G(x)=x^2g(x)-x h(g(x))AAx in Rdot Which of the following is/are correct? G^(prime)(1)=2 b. G^(prime)(1)=3 c. G^(1)=2 d. G^(1)=3

Suppose f(x)=e^(a x)+e^(b x) , where a!=b , and that f prime prime(x)-2f^(prime)(x)-15f(x)=0 for all xdot Then the value of (|a b|)/3 is ___

If function f satisfies the relation f(x)xf^(prime)(-x)=f(-x)xf^(prime)(x)fora l lx ,a n df(0)=3,a n diff(3)=3, then the value of f(-3) is ______________

f(x)=x^x , x in (0,oo) and let g(x) be inverse of f(x) , then g(x)' must be