If `f(1)=3,f^(prime)(1)=2,f ^(' ')(1)=4,t h e n(f^(-1))^(' ')(3)=` a. 1 b. `-1/2` c. -2 d. none of these

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^(prime)(x)=sqrt(2x^2-1) and y=f(x^2),t h e n(dy)/(dx) at x=1 is (a)2 (b) 1 (c) -2 (d) none of these

Let f(x) be a polynomial of degree 3 such that f(3)=1,f^(prime)(3)=-1,f^('')(3)=0,a n df^(''')(3)=12. Then the value of f^(prime)(1) is (a) 12 (b) 23 (c) -13 (d) none of these

If f(a)=2,f^(prime)(a)=1,g(a)=-1,g^(prime)(a)=2, then the value of lim_(x->a)(g(x)f(a)-g(a)f(x))/(x-a) is (a) -5 (b) 1/5 (c) 5 (d) none of these

ifint(dx)/(x^2(x^n+1)^(((n-1)/n)) )=-[f(x)]^(1/n)+c ,t h e nf(x)i s (a) (1+x^n) (b) 1+x^(-1) (c) x^n+x^(-n) (d) none of these

If f(x)=int_0^1(dt)/(1+|x-t|) ,then f^(prime)(1/2) is equal to (a)0 (b) 1/2 (c) 1 (d) none of these

If f(x)a n dg(x) are differentiable functions for 0lt=xlt=1 such that f(0)=10 ,g(0)=2,f(1)=2,g(1)=4, then in the interval (0,1)dot (a) f^(prime)(x)=0 for a l lx (b) f^(prime)(x)+4g^(prime)(x)=0 for at least one x (c) f(x)=2g'(x) for at most one x (d) none of these

A function f: R -> R satisfies the equation f(x)f(y)-f(x y)=x+yAAx ,y in Ra n df(1)>0 , then a. f(x)f^(-1)(x)=x^2-4 b. f(x)f^(-1)(x)=x^2-6 c. f(x)f^(-1)(x)=x^2-1 d. none of these

ifint(dx)/(x^2(x^n+1)^(((n-1)/n))=-[f(x)]^(1/n)+c ,t h e nf(x)i s (1+x^n) (b) 1+x^(-1) x^n+x^(-n) (d) none of these

f(x)=(1n(x^2+e^x))/(1n(x^4+e^(2x)))dotT h e n lim_(x->oo)f(x) is equal to (a) 1 (b) 1/2 (c) 2 (d) none of these