Maxima and Minima Formula

In this topic we will learn important maxima and minima formula for JEE Mains and Advanced and also important for Class 12 board student. So lets explore important formula of maxima and minima. 

MAXIMA-MINIMA

1. INTRODUCTION : MAXIMA AND MINIMA:

(a) Local Maxima /Relative maxima :

A function f ( x ) is said to have a local maxima at x = a if f ( a ) ≥ f ( x ) ∀ x ∈ ( a − h , a + h ) ∩ D_f(x) Where h is some positive real number.

(b) Local Minima/Relative minima:

A function f ( x ) is said to have a local minima at x = a if f ( a ) ≤ f ( x ) ∀ x ∈ ( a − h , a + h ) ∩ D_f(x) Where h is some positive real number.

(c) Absolute maxima (Global maxima):

A function f has an absolute maxima (or global maxima) at c if f ( c ) ≥ f ( x ) for all x in D , where D is the domain of f . The number f ( c ) is called the maximum value of f on D .

(d) Absolute minima (Global minima):

A function f has an absolute minima at c if f ( c ) ≤ f ( x ) for all x in D and the number f ( c ) is called the minimum value of f on D .

Note :

(i) The term 'extrema' is used for both maxima or minima.

(ii) A local maximum (minimum) value of a function may not be the greatest (least) value in a finite interval.

(iii) A function can have several extreme values such that local minimum value may be greater than a local maximum value.

(iv) It is not necessary that f ( x ) always has local maxima/minima at end points of the given interval when they are included.

2. DERIVATIVE TEST FOR ASCERTAINING MAXIMA AND MINIMA:

(a) First derivative test:

If f ′ ( x ) = 0 at a point ( say x = a ) and

(i) If f ′ ( x ) changes sign from positive to negative in the neighborhood of x = a then x = a is said to be a point local maxima.

(ii) If f ′ ( x ) changes sign from negative to positive in the neighborhood of x = a then x = a is said to be a point local minima.

Note : If f ′ ( x ) does not change sign i.e. has the same sign in a certain complete neighborhood of a, then f ( x ) is either increasing or decreasing throughout this neighborhood implying that x = a is not a point of extremum of f .

(b) Second derivative test:

If f ( x ) is continuous and differentiable at x = a where f ′ ( a ) = 0 (stationary points) and f ′ ′ ( a ) also exists then for ascertaining maxima/minima at x = a , 2^nd derivative test can be used -

(i) If f ′ ′ ( a ) > 0 ⇒ x = a is a point of local minima

(ii) If f ′ ′ ( a ) < 0 ⇒ x = a is a point of local maxima

(iii) If f ′ ′ ( a ) = 0 ⇒ second derivative test fails. To identify maxima/minima at this point either first derivative test or higher derivative test can be used.

(c) n^th derivative test:

Let f ( x ) be a function such that f ′ ( a ) = f ′ ′ ( a ) = f ′ ′ ( a ) = … = f^n-1 ( a ) = 0 & fⁿ ( a ) ≠ 0 , then

(i) If n is even & \(\left\{\begin{array}{c}\mathrm{f^n}(\mathrm{a})>0 \Rightarrow \text { Minima } \\ \mathrm{f}^{\mathrm{n}}(\mathrm{a})<0 \Rightarrow \text { Maxima }\end{array}\right.\)

(ii) If n is odd then neither maxima nor minima at x = a .

3. USEFUL FORMULAE OF MENSURATION TO REMEMBER- MAXIMA AND MINIMA

(a) Volume of a cuboid = ℓbh.

(b) Surface area of a cuboid = 2 ( ℓ b + b h + h ℓ ) .

(c) Volume of a prism = area of the base × height.

(d) Lateral surface area of prism = perimeter of the base × height.

(e) Total surface area of a prism = lateral surface area + 2 ⋅ area of the base (Note that lateral surfaces of a prism are all rectangles).

(f) Volume of a pyramid = 1/3 area of the base × height.

(g) Curved surface area of a pyramid = 1/2 (perimeter of the base) × slant height. (Note that slant surfaces of a pyramid are triangles).

(h) Volume of a cone = 1/3 π r² h .

(i) Curved surface area of a cylinder = 2 π r h .

(j) Total surface area of a cylinder = 2 π r h + 2 π r² .

(k) Volume of a sphere = 4/3 π r³ .

(l) Surface area of a sphere = 4 π r² .

(m) Area of a circular sector = 1/2 r² θ , when θ is in radians.

(n) Perimeter of circular sector = 2 r + r θ .

4. SIGNIFICANCE OF THE SIGN OF 2^ND ORDER DERIVATIVE - MAXIMA AND MINIMA

The sign of the 2^nd order derivative determines the concavity of the curve. i.e. If f ′ ′ ( x ) ≥ 0 ∀ x ∈ ( a , b ) then graph of f ( x ) is concave upward in ( a , b ) . Similarly if f ′ ′ ( x ) ≤ 0 ∀ x ∈ ( a , b ) then graph of f ( x ) is concave downward in (a, b).

5. SOME SPECIAL POINTS ON A CURVE - MAXIMA AND MINIMA

(a) Stationary points: The stationary points are the points of domain where f ′ ( x ) = 0 .

(b) Critical points : There are three kinds of critical points as follows:

    (i) The point at which f ′ ( x ) = 0

    (ii) The point at which f ′ ( x ) does not exists

    (iii) The end points of interval (if included) These points belongs to domain of the function.

Note : Local maxima and local minima occurs at critical points only but not all critical points will correspond to local maxima/ local minima.

(c) Point of inflection:

Point of inflection A point where the graph of a function has a tangent line and where the strict concavity changes is called a point of inflection. For finding point of inflection of any function, compute the points (x-coordinate) where d²y/dx² = 0 or d²y/dx² does not exist. Let the solution is x = a , if d²y/dx² = 0 at x = a and sign of d²y/dx² changes about this point then it is called point of inflection. If d²y/dx² does not exist at x = a and sign of d²y/dx² changes about this point and tangent exist at this point then it is called point of inflection.

6. SOME STANDARD RESULTS :

(a) Rectangle of largest area inscribed in a circle is a square.

(b) The function y = sin^mx cosⁿx attains the max value at \(x=\tan ^{-1} \sqrt{\dfrac{m}{n}}\)

(c) If 0 < a < b then | x − a | + | x − b | ≥ b − a and equality hold when x ∈ [ a , b ]

     If 0 < a < b < c then | x − a | + | x − b | + | x − c | ≥ c − a and equality hold when x = b

     If 0 < a < b < c < d then | x − a | + | x − b | + | x − c | + | x − d | ≥ d − a and equality hold when x ∈ [ b , c ] .

7. LEAST/GREATEST DISTANCE BETWEEN TWO CURVES :

Least/Greatest distance between two non-intersecting curves usually lies along the common normal (Wherever defined)

Note : Given a fixed point A ( a , b ) and a moving point P ( x , f ( x ) ) on the curve y = f ( x ) . Then A P will be maximum or minimum if it is normal to the curve at P .

Mathematics-Important Notes, concept & Formula

FAQs

What is Local Maxima or Relative Maxima?

A function f(x)is said to have a local maxima at x=a if f(a)≥f(x) ∀ ∈(a−h,a)∩ Domain of f(x) Where h is some positive real number

What is Local Minima or Relative minima?

A function f(x) is said to have a local minima at x=a if f(a)≤f(x) ∀ x∈(a−h,a+h)∩ Domain of f(x) Where h is some positive real number.

What is Absolute maxima (Global maxima)?

A function f has an absolute maxima (or global maxima) at c if f ( c ) ≥ f ( x ) for all x in D , where D is the domain of f . The number f ( c ) is called the maximum value of f on D .

What is Absolute minima (Global minima)?

A function f has an absolute minima at c if f ( c ) ≤ f ( x ) for all x in D and the number f ( c ) is called the minimum value of f on D .

Why first derivative test used?

Suppose we want to find maximum and minimum value of a function, in that case first derivative test used. If f ′ ( x ) = 0 at a point ( say x = a ) and
(i) If f ′ ( x ) changes sign from positive to negative in the neighbourhood of x = a then x = a is said to be a point local maxima.
(ii) If f ′ ( x ) changes sign from negative to positive in the neighbourhood of x = a then x = a is said to be a point local minima.

When and why we use second derivative test?

If f ( x ) is continuous and differentiable at x = a where f ′ ( a ) = 0 (stationary points) and f ′ ′ ( a ) also exists then for ascertaining maxima/minima at x = a , 2 nd derivative test can be used -

(i) If f ′ ′ ( a ) > 0 ⇒ x = a is a point of local minima

(ii) If f ′ ′ ( a ) < 0 ⇒ x = a is a point of local maxima

(iii) If f ′ ′ ( a ) = 0 ⇒ second derivative test fails.

To identify maxima/minima at this point either first derivative test or higher derivative test can be used.

When we use 'n' th derivative test ?

Let f ( x ) be a function such that f ′ ( a ) = f ′ ′ ( a ) = f ′ ′ ( a ) = … = f^( n − 1) ( a ) = 0 & f^ n ( a ) ≠ 0 , then 

 (i) If n is even & { f^ n ( a ) > 0 ⇒ Minima 

                             { f^ n ( a ) < 0 ⇒ Maxima 

 (ii) If n is odd then neither maxima nor minima at x = a

What is the important formula of mensuration to remember?

USEFUL FORMULAE OF MENSURATION TO REMEMBER

(a) Volume of a cuboid = ℓbh. 

(b) Surface area of a cuboid = 2 ( ℓ b + b h + h ℓ ) .

(c) Volume of a prism = area of the base × height.

(d) Lateral surface area of prism = perimeter of the base × height.

(e) Total surface area of a prism = lateral surface area + 2 ⋅ area of the base (Note that lateral surfaces of a prism are all rectangles).

(f) Volume of a pyramid = 1 /3 area of the base × height.

(g) Curved surface area of a pyramid = 1/2 (perimeter of the base) × slant height. (Note that slant surfaces of a pyramid are triangles).

(h) Volume of a cone = 1/3 π r^2 h .

(i) Curved surface area of a cylinder = 2 π r h .

(j) Total surface area of a cylinder = 2 π r h + 2 π r^2 .

(k) Volume of a sphere = 4/3 π r^3 .

(l) Surface area of a sphere = 4 π r^2 .

(m) Area of a circular sector = 1/2 r^2 θ , when θ is in radians.

(n) Perimeter of circular sector = 2 r + r θ .

What is the significant of the sign of 2nd derivative test?

The sign of the 2 nd order derivative determines the concavity of the curve. i.e. If f ′ ′ ( x ) ≥ 0 ∀ x ∈ ( a , b ) then graph of f ( x ) is concave upward in ( a , b ) . Similarly if f ′ ′ ( x ) ≤ 0 ∀ x ∈ ( a , b ) then graph of f ( x ) is concave downward in (a, b).

Where we find stationary point on the curve?

Stationary points: The stationary points are the points of domain where f ′ ( x ) = 0 .

What are the three kinds of critical point?

Critical points : There are three kinds of critical points as follows:

(i) The point at which f ′ ( x ) = 0

(ii) The point at which f ′ ( x ) does not exists

(iii) The end points of interval (if included) These points belongs to domain of the function.

What is point of inflection ?

A point where the graph of a function has a tangent line and where the strict concavity changes is called a point of inflection.

How do you find the point of inflection?

For finding point of inflection of any function, compute the points (x-coordinate) where d^2 y/ d x^2 = 0 or d^2 y / d x^2 does not exist. Let the solution is x = a , if d^2 y/ d x^2 = 0 at x = a and sign of d^2 y/ d x^2 changes about this point then it is called point of inflection.

If d^2 y/ d x^2 does not exist at x = a and sign of d^2 y/ d x^2 changes about this point and tangent exist at this point then it is called point of inflection.

What is a rectangle of largest area inscribed in a circle ?

Rectangle of largest area inscribed in a circle is a square.

What is the least/greatest distance between two curves ?

Least/Greatest distance between two non-intersecting curves usually lies along the common normal (Wherever defined)