The `n t h` derivative of the function `f(x)=1/(1-x^2)[w h e r ex in (-1,1)]` at the point `x=0` where `n` is even is (a) 0 (b) `n !` (c) `n^n C_2` (d) `2^n C_2`

The n t h derivative of x e^x vanishes when (a) x=0 (b) x=-1 (c) x=-n (d) x=n

If t a n y=(2^x)/(1+2^(2x+1)),t h e n(dy)/(dx) at x=0 is

The derivative of y=(1-x)(2-x)....(n-x) at x=1 is (a) 0 (b) (-1)(n-1)! (c) n !-1 (d) (-1)^(n-1)(n-1)!

If y=a x^(n+1)+b x^(-n),t h e nx^2(d^2y)/(dx^2) is equal to (a) n(n-1)y (b) n(n+1)y (c) n y (d) n^2y

If the expansion in powers of x of the function 1/[(1-a x)(1-b x)] is a a_0+a_1x+a_2x^2+a_3x^3+ ,t h e na_n i s a. (b^n-a^n)/(b-a) b. (a^n-b^n)/(b-a) c. (b^(n+1)-a^(n+1))/(b-a) d. (a^(n+1)-b^(n+1))/(b-a)

f(x)=(1n(x^2+e^x))/(1n(x^4+e^(2x)))dotT h e n lim_(x->oo)f(x) is equal to (a) 1 (b) 1/2 (c) 2 (d) none of these

Which of the following function is/are periodic? A. f(x)={1,0,x is rationalx is irrational B. f(x)={x−[x],12,2n≤x<2n+12n+1≤x<2n+2 where [.] denotes the greatest integer function n∈Z C. f(x)=(−1)[2xπ, where [.] denotes the greatest integer function D. f(x)=x−[x+3]+tan(πx2), where [.] denotes the greatest integer function, and a is a rational number

The function f(x)=(4sin^2x−1)^n (x^2−x+1),n in N , has a local minimum at x=pi/6 . Then (a) n is any even number (b) n is an odd number (c) n is odd prime number (d) n is any natural number

If f^(prime)(x)=sqrt(2x^2-1) and y=f(x^2),t h e n(dy)/(dx) at x=1 is (a)2 (b) 1 (c) -2 (d) none of these

The function f(x)=(sec^(-1)x)/(sqrt(x-[x]), where [x] denotes the greatest integer less than or equal to x , is defined for all x in (a) R (b) R-{(-1,1)uu{n"|"n in Z}} (c) R-(0,1) (d) R-{n|n in N}