If `f ''(x)|le 1 AA x in R, and f (0) =0=f' (0),` then which of the following can not be true ?

Let f (x) =ax ^(2) + bx+ c,a gt = and f (2-x) =f (2+x) AA x in R and f (x) =0 has 2 distinct real roots, then which of the following is true ?

If (f(x)-1) (x ^(2) + x+1)^(2) -(f (x)+1) (x^(4) +x ^(2) +1) =0 AA x in R -{0}and f (x) ne pm 1, then which of the following statement (s) is/are correct ?

If f''(x) le0 "for all" x in (a,b) then f'(x)=0

Let f'(sin x)lt0 and f''(sin x) gt0 forall x in (0,(pi)/(2)) and g(x) =f(sinx)+f(cosx) which of the following is true?

Let f(x+y)+f(x-y)=2f(x)f(y) AA x,y in R and f(0)=k , then

f(x)={(x-1",",-1 le x le 0),(x^(2)",",0le x le 1):} and g(x)=sinx Consider the functions h_(1)(x)=f(|g(x)|) and h_(2)(x)=|f(g(x))|. Which of the following is not true about h_(2)(x) ?

If f:R rarr (0,oo) be a differentiable function f(x) satisfying f(x+y)-f(x-y)=f(x)*{f(y)-f(-y)}, AA x, y in R, (f(y)!= f(-y) " for all " y in R) and f'(0)=2010 . Now, answer the following questions. Which of the following is true for f(x)

A function f(x) having the following properties, (i) f(x) is continuous except at x=3 (ii) f(x) is differentiable except at x=-2 and x=3 (iii) f(0) =0 lim_(x to 3) f(x) to - oo lim_(x to oo) f(x) =3 , lim_(x to oo) f(x)=0 (iv) f'(x) gt 0 AA in (-oo, -2) uu (3,oo) " and " f'(x) le 0 AA x in (-2,3) (v) f''(x) gt 0 AA x in (-oo,-2) uu (-2,0)" and "f''(x) lt 0 AA x in (0,3) uu(3,oo) Then answer the following questions Find the Maximum possible number of solutions of f(x)=|x|

Let |f (x)| le sin ^(2) x, AA x in R, then

If a function satisfies the relation f(x) f''(x)-f(x)f'(x)=(f'(x))^(2) AA x in R and f(0)=f'(0)=1, then The value of lim_(x to -oo) f(x) is