Class 12 Chapter 6 (Application Of Derivative) Class Notes part II

MAXIMA AND MINIMA

 

1. MAXIMA AND MINIMA:

   • Let f be a function defined on an interval I. Then

     1. f is said to have a maximum value in I, if there exists a point c in I such that f (c) ≥ f (x) , for all x ∈ I.

   • The number f (c) is called the maximum value of f in I and the point c is called a point of maximum value of f in I.

     1. f is said to have a minimum value in I, if there exists a point c in I such that f (c) ≤ f (x), for all x ∈ I.

   • The number f (c), in this case, is called the minimum value of f in I and the point c, in this case, is called a point of minimum value of f in I.

   • f(x) is said to have an extreme value in I if there exists a point c in I such that f is either a maximum value or a minimum value of f in I.

   • The number f (c), in this case, is called an extreme value of f in I and the point c is called an extreme point.

     Absolute maxima and minima

   • Let f be a function defined on the interval I and c ∈ I. Then

     1. f(c) is absolute minimum if f(x) ≥ f(c) for all x ∈ I.

     2. f(c) is absolute maximum if f(x) ≤ f(c) for all x ∈ I.

     3. c ∈ I is called the critical point off if f ′(c) = 0

     4. Absolute maximum or minimum value of a continuous function f on [a, b] occurs at a or b or at critical points off (i.e. at the points where f ′is zero)

   • If c₁ ,c₂, … , c_n are the critical points lying in [a , b], then

     absolute maximum value of f = max{f(a), f(c₁), f(c₂), … , f(c_n), f(b)} and absolute minimum value of f = min{f(a), f(c₁), f(c₂), … , f(c_n), f(b)}.

     Local maxima and minima

   • (a)A function f is said to have a local maxima or simply a maximum value at x a if f(a ± h) ≤ f(a) for sufficiently small h

   • (b)A function f is said to have a local minima or simply a minimum value at x = a if f(a ± h) ≥ f(a).

   • First derivative test : A function f has a maximum at a point x = a if

     1. f ′(a) = 0, and

     2. f ′(x) changes sign from + ve to –ve in the neighborhood of ‘a’ (points taken from left to right).

     However, f has a minimum at x = a, if

     1. f ′(a) = 0, and

     2. f ′(x) changes sign from –ve to +ve in the neighborhood of ‘a’.

        If f ′(a) = 0 and f’(x) does not change sign, then f(x) has neither maximum nor minimum and the point ‘a’ is called point of inflection.

     
   • The points where f ′(x) = 0 are called stationary or critical points. The stationary points at which the function attains either maximum or minimum values are called extreme points.
   

   • Second derivative test :

     1. a function has a maxima at x= a, if f ′(x) =0 and f ′′ (a) <0

     2. a function has a minima at x = a, if f ′(x) = 0 and f ′′(a) > 0.

 

EXAMPLE

QUESTION: If length of three sides of a trapezium other than base is equal to 10cm each, then find the area of the trapezium when it is maximum.

SOLUTION: The required trapezium is as given in Fig below. Draw perpendiculars DP and CQ on AB

ΔAPD is congruent to ΔBQC

Let AP = BQ = x cm

\[ \Rightarrow DP = QC = \sqrt {100 - {x^2}} \]

Area of Trapezium = \( \Rightarrow \dfrac{1}{2}(2x + 10 + 10)\sqrt {100 - {x^2}}  = (x + 10)\sqrt {100 - {x^2}} \)

\[ \Rightarrow A'(x) = \dfrac{{ - 2{x^2} - 10x - 100}}{{\sqrt {100 - {x^2}} }}\]

\[ \Rightarrow A'(x) = 0 \Rightarrow x = 5\]

Now, \( \Rightarrow A''(x) = \dfrac{{2{x^2} - 300x - 100}}{{{{\left( {100 - {x^2}} \right)}^{\dfrac{2}{3}}}}}\)

and \( \Rightarrow {\left. {A''(x)} \right|_{x = 5}} = \dfrac{{ - 30}}{{{{\left( {75} \right)}^{\frac{2}{3}}}}} < 0\)

thus , the area of trapezium is maximum at x = 5

\[Area = 75\sqrt 3 \,c{m^2}\]

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