If `P(1)=0a n d(d P(x))/(dx)gtP(x)`, for all ` xge1`. Prove that `P(x)>0` for all `x>1.`

Show that x/((1+x))<1n(1+x) for x>0

Solve (x+1)/(x-1)gt0

If H(x_(0)) =0 for some x= x_(0) and (d)/(dx)H(x)gt2cxH(x) for all xgex_(0) where cgt0 then

Solve: (x+1)/(x-1) gt 0

Let O be the origin. If A(1,0)a n dB(0,1)a n dP(x , y) are points such that x y >0a n dx+y<1, then (a) P lies either inside the triangle O A B or in the third quadrant. (b) P cannot lie inside the triangle O A B (c) P lies inside the triangle O A B (d) P lies in the first quadrant only

If f(a-x)=f(a+x) " and " f(b-x)=f(b+x) for all real x, where a, b (a gt b gt 0) are constants, then prove that f(x) is a periodic function.

f(x)=e^(-1/x),w h e r ex >0, Let for each positive integer n ,P_n be the polynomial such that (d^nf(x))/(dx^n)=P_n(1/x)e^(-1/x) for all x > 0. Show that P_(n+1)(x)=x^2[P_n(x)-d/(dx)P_n(x)]

Suppose p(x)=a_0+a_1x+a_2x^2++a_n x^ndot If |p(x)|lt=e^(x-1)-1| for all xgeq0, prove that |a_1+2a_2++n a_n|lt=1.

Prove that (1+x)^(n) ge (1+nx) for all natural number n where x gt -1

Let f be a real-valued function defined on interval (0,oo) ,by f(x)=lnx+int_0^xsqrt(1+sint).dt . Then which of the following statement(s) is (are) true? (A). f"(x) exists for all in (0,oo) . " " (B). f'(x) exists for all x in (0,oo) and f' is continuous on (0,oo) , but not differentiable on (0,oo) . " " (C). there exists alpha>1 such that |f'(x)|<|f(x)| for all x in (alpha,oo) . " " (D). there exists beta>1 such that |f(x)|+|f'(x)|<=beta for all x in (0,oo) .