MAXIMA AND MINIMA

MAXIMA AND MINIMA

1. MAXIMA

   • Let f(x) be a real function defined on an interval [a ,b].
     • MINIMA
       • then there exist a point c in [a,b] such that f(x)>f(c) for all x[a,b]
   • then there exist a point c in [a,b] such that f(x)<f(c) for all x[a,b]

2. LOCAL MAXIMA AND MINIMA

   • LOCAL MAXIMA
     • A function f(x) attain a local maxima at x=a if there exist a neighbourhood (a -δ, a+δ) of a such that
       • f(x)<f(a) for all x(a -δ, a+δ) ,x=a
         • f(x)-f(a) <0for all x(a -δ, a+δ) ,x=a .f(a) is the local maxima
   • LOCAL MINIMA
     • A function f(x) attain a local maxima at x=a if there exist a neighbourhood (a -δ, a+δ) of a such that
       • f(x)>f(a) for all x(a -δ, a+δ) ,x=a
         • f(x)-f(a) >0for all x(a -δ, a+δ) ,x=a .f(a) is the local minima

3. FIRST DERIVATIVE TEST FOR LOCAL MAXIMA AND MNIMA

   • Put y=f(x) and find dy/dx
     • Put dy/dx =0 and solve this equation for x. Let C1,C2........Cn be the roots of the equation .these point are stationary or critical points of f(x) where the function can attain local maxima or minima .So we test for each of these points.
       • Consider x=C1
         • Continu this for other values of x
         • If dy/dx does not change its sign as x increases through c1 then x=C1 is neither a maximum nor a point of local minimum .This point is called POINT OF INFLEXION
         • If dy/dx changes its sign from - to + as x increases through c1 then fuction attains a local minimum at x=C1
         • If dy/dx changes its sign from + to - as x increases through c1 then fuction attains a local maximum at x=C1

4. HIGHER ORDER DERIVATVE TEST

   • Find f'(x)
     • Put f'(x) =0 and solve this equation for x. Let C1,C2........Cn be the roots of the equation .these point are stationary or critical points of f(x) where the function can attain local maxima or minima .So we test for each of these points.
       • Find f ''(x). Consider x=C1
         • If f ''(C1) < 0,then x=C1 is a point of local maximum
         • If f ''(C1) > 0,then x=C1 is a point of local minimum
         • if f ''(C1) =0 we must find f '''(x) and substitue C1 in it.
           • If f '''(C1)=0,then x=C1 is neither a point of local maxima or minima and is called point of inflection
             • If f '''(C1)=0 . find f'''' (x) and so on

5. MAXIMUM AND MINIMUM VALUES IN A CLOSED INTERNAL

   • Find f'(x)
     • Put f'(x)=0 and find values of x .Let C1,C2...........Cn be values of x
       • Take the maximum and minimum values out of the values f(a),f(C1),f(C2)......f(Cn),f(b). The max and min values obtained are respectively the absolute maximum and minimum values of function .

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