Let `g(x)=f(logx)+f(2-logx)a n df^(x)<0AAx in (0,3)dot` Then find the interval in which `g(x)` increases.

Let g(x)=2f(x/2)+f(2-x)a n df^('')(x)<0AAx in (0,2)dot Then g(x) increases in (a) (1/2,2) (b) (4/3,2) (c) (0,2) (d) (0,4/3)

Let g(x)=e^(f(x))a n df(x+1)=x+f(x)AAx in Rdot If n in I^+,t h e n(g^(prime)(n+1/2))/(g(n+1/2))-(g^(prime)(1/2))/(g(1/2))= 2(1+1/2+1/3++1/n) 2(1+1/3+1/5+1/(2n-1)) n 1

Let f(x)=x+f(x-1)forAAx in RdotIff(0)=1,fin df(100)dot

Which of the following function is non-differentiable in domain? f(x)=(x-2)/(x^2+3) (b) f(x)=log|x| f(x)=x^3logx (d) f(x)=(x-3)^(3/5)

Iff(2-x)=f(2+x)a n df(4-x)=f(4+x) for all xa n df(x) is a function for which int_0^2f(x)dx=5,t h e nint_0^(50)f(x)dx is equal to 125 (b) int_(-4)^(46)f(x)dx int_1^(51)f(x)dx (d) int_2^(52)f(x)dx

If x^y=e^(x-y), Prove that (dy)/(dx)=(logx)/((1+logx)^2)

Let f(x+y)=f(x)dotf(y) for all xa n dydot Suppose f(5)=2a n df^(prime)(0)=3. Find f^(prime)(5)dot

Let f(x y)=f(x)f(y)AAx , y in Ra n df is differentiable at x=1 such that f^(prime)(1)=1. Also, f(1)!=0,f(2)=3. Then find f^(prime)(2)dot

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))

f(x)=tan^(-1){log(e/x^2)/log(ex^2)}+tan^(-1)((3+2logx)/(1-6logx)) then find (d^ny)/(dx^n)