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)=2 and y(x)=f(e^x)e^(f(x)) , then y^(prime)(0) is equal to a. 1 b. 2 c. 0 d. none of these

If f(x)=e^xg(2x),g(0)=4 and g'(0) = 2 then f'(0) is equal to

Let f: R -> R be a differentiable function such that f(0) = 0,f(pi/2)=3a n df^(prime)(0)=1. If g(x)=int_x^(pi/2)[f^(prime)(t)cos e ct-cottcos e ctf(t)]dt for x(0,pi/2], then (lim)_(x -> 0)g(x)=

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 f(x) be differentiable for real such that f'(x)lt0" on "(-4,6) and f(x) gt 0" on "(6,oo). If g(x)=f(10 - 2x) , then the value of g'(2) is

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 ___

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

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___

Let g(x)=2f(x/2)+f(x) and f''(x)lt0 " in "0lexle1 , then g(x)

f(x)=e^x.g(x) , g(0)=2 and g'(0)=1 then f'(0)=