This is a basic knowledge of Maxima-Minima which is sometimes useful to solve some problems of IIT Entrance Exams.

Basic Knowledge of Maxima-Minima:-
<i> A function f(x) is said to have maxima at x=a if f(x)≤f(a) for x belongs to (a-k,a+k) and minima at x=b if f(x)≥f(b) for x belongs to (b-k,b+k), for some k>0.
<ii> If the function f(x) is differentiable at a point x=a, then f'(a)=0 is the necessary but not sufficient for extremum values.
[ e.g.  f(x)=x^3 , f'(x)=0 for x=0 but it has no extremum at x=0. ]

Problem:-[IIT-JAM:MS-2009] Let f be a differentiable function defined on [0,1] . If  ζ∈(0,1) is such that f(x)<f(ζ)=f(0) for all x is in (0,1], x≠ζ, then 
(a) f'(ζ)=0 and f"(0)=0                    (b) f'(ζ)=0 and f"(ζ)≤0
(c) f'(ζ)>0 and f"(0)≤0                    (d) f'(x)=0 and f"(0)>0  
Sol:-  (b) As f(x)<f(ζ) so there exists a local minima at x=ζ for all x in (0,1]
=>  f'(ζ)=0 and as f(ζ)=f(0) is maximum.
So, f"(ζ)≤ 0
i.e. f"(0)≤ 0.