Let `f: RvecR` be such that `f(a)=1,f^(prime)(a)=2.` Then `(lim)_(xvec0)((f^2(a+x))/(f(a)))^(1//x)` is a. `e^2` b. `e^4` c. `e^(-4)` d. `1//e`

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

f(x)=(e^(1/x)-1)/(e^(1/x)+1) find f^(-1)(x)

If f(x)=e^(px+q) ,then shoq that f(a).f(b).f(c)=f(a+b+c).e^(2q)

f(x)={x ,xlt=0 1,x=0,t h e nfin d("lim")_(xvec0)f(x)x^2,x >0ife xi s t s

Let ("lim")_(xvec1)(x^a-a x+a-1)/((x-1)^2)=f(a)dot Then the value of f(4) is _________

If f(x)=e^(px+q) , then show that f(a).f(b).f(c)=f(a+b+c).e^(2q)

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

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(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

Let f(x) be a non-constant twice differentiable function defined on (-oo,oo) such that f(x)=f(1-x)a n df^(prime)(1/4)=0. Then (a) f^(prime)(x) vanishes at least twice on [0,1] (b) f^(prime)(1/2)=0 (c) int_(-1/2)^(1/2)f(x+1/2)sinxdx=0 (d) int_(-1/2)^(1/2)f(t)e^(sinpit)dt=int_(1/2)^1f(1-t)e^(sinpit)dt