Show that `ln\ (1+x)> x-(x^2)/2\ AA\ x\ in (0,oo)`

Show that tan^(-1)x>(x)/(1+(x^(2))/(3)) if x in(0,oo)

Show that (d^n)/(dx^(n) )(x^(n) log x) = n! (log x + 1+(1)/(2) +…+(1)/(n)) AA n in N .

Show that 1 + x log (x + sqrt (x ^(2) + 1)) ge sqrt ( 1 + x ^(2)) AA x ge 0

Let f(x) lt 0 AA x in (-=oo, 0) and f (x) gt 0 AA x in (0,oo) also f (0)=o, Again f'(x) lt 0 AA x in (-oo, -1) and f '(x) gt AA x in (-1,oo) also f '(-1)=0 given lim _(x to oo) f (x)=0 and lim _(x to oo) f (x)=oo and function is twice differentiable. If f'(x) lt 0 AA x in (0,oo)and f'(0)=1 then number of solutions of equatin f (x)=x ^(2) is :

Let f(x) lt 0 AA x in (-oo, 0) and f (x) gt 0 AA x in (0,oo) also f (0)=0, Again f'(x) lt 0 AA x in (-oo, -1) and f '(x) gt AA x in (-1,oo) also f '(-1)=0 given lim _(x to oo) f (x)=0 and lim _(x to oo) f (x)=oo and function is twice differentiable. The minimum number of points where f'(x) is zero is:

Show that 0

Find f(x) when it is given by f(x)=max{x^(3),x^(2),(1)/(64)},AA x in [0, oo).

The function f(x)=log x-(2x)/(x+2) is increasing for all A) x in(-oo,0) , B) x in(-oo,1) C) x in(-1,oo) D) x in(0,oo)

The range of (x ^ (2) -2x + 3) / (x ^ (2) -2x-8) AA x in R is (i) (- oo, 0] uu [1, oo) (ii) [ (1) / (2), 2] (iii) (- oo, - (2) / (9)] uu [1, oo) (iv) (oo, -6] uu ([- 2, oo)