Let `f: RvecR` be a continuous and differentiable function such that `(f(x^2+1))^(sqrt(x))=5forAAx in (0,oo),` then the value of `(f((16+y^2)/(y^2)))^(4/(sqrt(y)))fore a c hy in (0,oo)` is equal to (a) 5 (b) 25 (c) 125 (d) 625

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:(0,oo)->R be a differentiable function such that f'(x)=2-f(x)/x for all x in (0,oo) and f(1)=1 , then

Suppose f: RvecR^+ be a differentiable function such that 3f(x+y)=f(x)f(y)AAx ,y in R with f(1)=6. Then the value of f(2) is 6 b. 9 c. 12 d. 15

Let f:[0,oo)vecR be a continuous strictly increasing function, such that f^3(x)=int_0^x tdotf^2(t)dt for every xgeq0. Then value of f(6) is_______

Let f:[2,7]vec[0,oo) be a continuous and differentiable function. Then show that (f(7)-f(2))((f(7))^2+(f(2))^2+f(2)f(7))/3=5f^2(c)f^(prime)(c), where c in [2,7]dot

Let f(x) be a non-constant twice differentiable function defined on (oo, oo) such that f(x) = f(1-x) and f"(1/4) = 0 . Then

Consider a differentiable f:R to R for which f(1)=2 and f(x+y)=2^(x)f(y)+4^(y)f(x) AA x , y in R. The value of f(4) is

Let f be a differentiable function from R to R such that abs(f(x)-f(y))abs(le2)abs(x-y)^(3//2) ,for all x,y inR .If f(0)=1 ,then int_(0)^(1)f^2(x)dx is equal to

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

Let f:[0,2]->R be a function which is continuous on [0,2] and is differentiable on (0,2) with f(0)=1 L e t :F(x)=int_0^(x^2)f(sqrt(t))dtforx in [0,2]dotIfF^(prime)(x)=f^(prime)(x) . for all x in (0,2), then F(2) equals (a) e^2-1 (b) e^4-1 (c) e-1 (d) e^4