Let `f: R->R` be such that `f(1)=3a n df^(prime)(1)=6.` Then `lim_(x->0)((f(1+x))/(f(1)))^(1//x)=` (a)`1` (b) `e^(1/2)` (c) `e^2` (d) `e^3`

Let f:RrarrR be such that f(a)=1, f(a)=2 . Then lim_(x to 0)((f^(2)(a+x))/(f(a)))^(1//x) is

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 f: RvecR be a one-one onto differentiable function, such that f(2)=1a n df^(prime)(2)=3. The find the value of ((d/(dx)(f^(-1)(x))))_(x=1)

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

Let f(x) be the fourth degree polynomial such that f^(prime)(0)=-6,f(0)=2 and (lim)_(x->1)(f(x))/((x-1)^2)=1 The value of f(2) is a. 3 b. 1 c. 0 d. 2

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: RvecR be a differen tiable function atg x=0 satisfying f(0)=0a n df^(prime)(0)=1 , then the value of (lim)_(xvec0)1/xsum_(n=1)^oo(-1)^nf(x/n)= e b. -log2 c. 0 d. 1

Compute f'(0^(+)) if f(x)=(x(e^(1/x)-1))/(e^(1/x)+1) :

Let f(1)=-2a n df^(prime)(x)geq4. 2for1lt=xlt=6. The smallest possible value of f(6) is 9 (b) 12 (c) 15 (d) 19

If f(x)=sin^(-1)x then prove that lim_(x->1/2)f(3x-4x^3)=pi-3lim_(x->1/2)sin^(-1)x