`f:R->R, f(x)` is a differentiable bijective function, then which of the following is true?
(a) `(f(x)-x) f''(x) lt 0`
(b) `(f(x)-x) f''(x) gt 0`
(c) `(f(x)-x) f''(x) gt 0` then `f(x)=f^-1(x)` has no solution
(d) if `(f(x)-x) f''(x) gt 0` then `f(x)=f^-1(x)`

If f(x)=a^(x), a gt 0 , then f is

If f(x)gt0 and differentiable in R, then : f'(x)=

If f(x) is a differentiable real valued function such that f(0)=0 and f\'(x)+2f(x) le 1 , then (A) f(x) gt 1/2 (B) f(x) ge 0 (C) f(x) le 1/2 (D) none of these

If f(x) is a differentiable real valued function such that f(0)=0 and f\'(x)+2f(x) le 1 , then (A) f(x) gt 1/2 (B) f(x) ge 0 (C) f(x) le 1/2 (D) none of these

If f(x) is a differentiable real valued function satisfying f''(x)-3f'(x) gt 3 AA x ge 0 and f'(0)=-1, then f(x)+x AA x gt 0 is

If f(x) is a differentiable real valued function satisfying f''(x)-3f'(x) gt 3 AA x ge 0 and f'(0)=-1, then f(x)+x AA x gt 0 is

If f(x) is a differentiable real valued function satisfying f''(x)-3f'(x) gt 3 AA x ge 0 and f'(0)=-1, then f(x)+x AA x gt 0 is

If f(x) is twice differentiable and continuous function in x in [a,b] also f'(x) gt 0 and f ''(x) lt 0 and c in (a,b) then (f(c) - f(a))/(f(b) - f(a)) is greater than

Let f:[0,1]rarrR (the set of all real numbers) be a function. Suppose the function f is twice differentiable, f(0)=f(1)=0 and satisfies f\'\'(x)-2f\'(x)+f(x) ge e^x, x in [0,1] If the function e^(-x)f(x) assumes its minimum in the interval [0,1] at x=1/4 , which of the following is true? (A) f\'(x) lt f(x), 1/4 lt x lt 3/4 (B) f\'(x) gt f(x), 0 lt x lt 1/4 (C) f\'(x) lt f(x), 0 lt x lt 1/4 (D) f\'(x) lt f(x), 3/4 lt x lt 1