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

If x gt 0 , prove that, x gt log (1+x) gt x/(1+x)

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.

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)]

If x gt 0 , show that, x gt log (x^(2)+1)

Prove that , int a^(x) dx = (a^(x))/(log_(e)a)+ c , where a gt 0 . Does the formula hold for a = 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) .

Solve (x-1 )(x-2)(1-2x) gt 0

If f(x)="max" (x,(1)/(x))" for " x gt 0 , where max (p,q) denotes the greater of p and q, find the value of f(a)f((1)/(a)) , wheres a gt 0