Let `f(x)=xsinpix ,x > 0.` Then for all natural numbers `n ,f^(prime)(x)` vanishes at (A)a unique point in the interval `(n , n+1/2)` (B)a unique point in the interval `(n+1/2, n+1)` (C)a unique point in the interval `(n , n+1)` (D)two points in the interval `(n , n+1)`

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

Let f((x+y)/2)=(f(x)+f(y))/2fora l lr e a lxa n dy If f^(prime)(0) exists and equals -1a n df(0)=1,t h e n f i n d f(2)dot

If f(x) =(p-x^n)^(1/n) , p >0 and n is a positive integer then f[f(x)] is equal to

Let f(x) = (x^2 - 1)^(n+1) + (x^2 + x + 1) . Then f(x) has local extremum at x = 1 , when n is (A) n = 2 (C) n = 4 (B) n = 3 (D) n = 5

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

The sum of coefficient in the expansion of (1+a x-2x^2)^n is (a)positive, when a<1a n dn=2k ,k in N (b)negative, when a<1a n dn=2k+1,k in N (c)positive, when a<1a n dn in N (d)zero, when a=1

If f(x)=x/(sinx)a n dg(x)=x/(tanx),w h e r e0ltxlt=1, then in this interval

Let f(x)=(x-1)^4(x-2)^n ,n in Ndot Then f(x) has a maximum at x=1 if n is odd a maximum at x=1 if n is even a minimum at x=1 if n is even a minima at x=2 if n is even

Let f be function f:N to N be defined by f(x)=3x+2, xin N . Find the images of 1, 2, 3

f(x) is cubic polynomial with f(x)=18a n df(1)=-1 . Also f(x) has local maxima at x=-1a n df^(prime)(x) has local minima at x=0 , then the distance between (-1,2)a n d(af(a)), where x=a is the point of local minima is 2sqrt(5) f(x) is increasing for x in [1,2sqrt(5]) f(x) has local minima at x=1 the value of f(0)=15