#### MATHS-TRICK-15

*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

**.**