A differentiable function f(x) satisfies `f(0)=0 and f(1)=sin1`, then (where f' represents derivative of f)

Suppose |(f'(x),f(x)),(f''(x),f'(x))|=0 where f(x) is continuous differentiable function with f'(x) !=0 and satisfies f(0)=1 and f'(0)=2 , then f(x)=e^(lambda x)+k , then lambda+k is equal to ..........

Suppose |(f'(x),f(x)),(f''(x),f'(x))|=0 where f(x) is continuous differentiable function with f'(x) !=0 and satisfies f(0)=1 and f'(0)=2 , then f(x)=e^(lambda x)+k , then lambda+k is equal to ..........

Suppose |(f'(x),f(x)),(f''(x),f'(x))|=0 where f(x) is continuously differentiable function with f'(x)ne0 and satisfies f(0) = 1 and f'(0) = 2 then lim_(xrarr0) (f(x)-1)/(x) is

Let f (x) be a differentiable function and f '(1) =4,f'(2) =6 where f '(c ) means the derivative of function f(x) at x=c. Then lim _(hto0)(f(2 +2h +h^(2))-f(2))/(f (1+h-h^(2))-f(1)) is equal to-

Suppose |[f'(x),f(x)],[f''(x),f'(x)]|=0 is continuously differentiable function with f^(prime)(x)!=0 and satisfies f(0)=1 and f'(0)=2 then (lim)_(x->0)(f(x)-1)/x is 1//2 b. 1 c. 2 d. 0

Suppose |[f'(x),f(x)],[f''(x),f'(x)]|=0 is continuously differentiable function with f^(prime)(x)!=0 and satisfies f(0)=1 and f'(0)=2 then (lim)_(x->0)(f(x)-1)/x is 1//2 b. 1 c. 2 d. 0

Suppose |[f'(x),f(x)],[f''(x),f'(x)]|=0 is continuously differentiable function with f^(prime)(x)!=0 and satisfies f(0)=1 and f'(0)=2 then lim_(x->0)(f(x)-1)/x is a. 1//2 b. 1 c. 2 d. 0