Let a > 0 and f be continuous in [-a, a]. Suppose that `f'(x)`exists and `f'(x)<= 1` for all `x in(-a,a). if f(a)=a and f(-a)=-a,` show that `f(0)=0.`

Let f be continuous on [a, b] and assume the second derivative f" exists on (a, b). Suppose that the graph of f and the line segment joining the point (a,f(a)) and (b,f(b)) intersect at a point (x_0,f(x_0)) where a < x_0 < b. Show that there exists a point c in(a,b) such that f"(c)=0.

Let f(x) be defined for all x > 0 and be continuous. Let f(x) satisfies f(x/y)=f(x)-f(y) for all x,y and f(e)=1. Then

Let f be continuous function on [0,oo) such that lim _(x to oo) (f(x)+ int _(o)^(x) f (t ) (dt)) exists. Find lim _(x to oo) f (x).

Let f(x) be a continuous function such that f(0)=1 and f(x)=f((x)/(7))=(x)/(7)AA x in R then f(42) is

Let f(x) be a continuous function such that f(0) = 1 and f(x)-f (x/7) = x/7 AA x in R , then f(42) is

Let f(x) be a continuous function such that f(0) = 1 and f(x)=f(x/7)=x/7 AA x in R, then f(42) is