# Maxima and Minima Formulas

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## Maxima and Minima Formulas List

You will find the below listed Maxima and Minima Formulae quite useful during your calculations. Instead of going with the traditional and lengthy methods to arrive at the solution try solving the Maxima and Minima related problems using Formula Sheet & Tables existing. Maxima and Minima Formulas List covers all kinds and you can solve both basic and advanced level problems easily.

1. Maximum & Minimum Points

Maxima:

A function f(x) is said to be maximum at x = a, if there j exists a very small positive number h, such that

f(x) < f(a) ∀ x ∈ (a – h, a + h), x ≠ a

Minima:

A function f(x) is said to be minimum at x = b, if there exists a very small positive number h, such that f(x) > f(b), ∀ x ∈ (b – h, b + h), x ≠ b

Note:

- The maximum and minimum points are also known as extreme points.
- A function may have more than one maximum and minimum points in an interval.
- A maximum value of a function f(x) in an interval [a, b] is not necessarily its greatest value in that interval. Similarly, a minimum value may not be the least value of the function.
- If a continuous function has only one maximum (minimum) point, then at this point function has its greatest (least) value.
- Monotonic functions do not have extreme points.

2. Conditions for Maxima & Minima of a function

A. Necessary condition:

A point x = a is an extreme point of a function f(x) if f'(a) = 0, provided f'(a) exists. Thus if f'(a) exists, then

x = a is an extreme point ⇒ f'(a) = 0
or f'(a) ≠ 0 ⇒ x = a is not an extreme point |

B. Sufficient condition:

- The value of the function f(x) at x = a is maximum, if f'(a) = 0 and f “(a) < 0.
- The value of the function f(x) at x = a in minimum, if f'(a) = 0 and f” (a) > 0.

Note:

(i) If f'(a) = 0, f”(a) = 0, f”‘(a) ≠ 0 then x = a is not an extreme point for the function f(x).

(ii) If f'(a) = 0, f”(a) = 0, f”‘(a) = 0 then the sign of f^{(iv)} (a) will determine the maximum and minimum value of function i.e.f(x) is maximum, if f^{(iv)} (a) < 0 and minimum if f^{(iv)} (a) > 0.

(iii) If f (a) = f”(a) = f”‘(a) = …….. = f^{n-1} (a) = 0 and f^{n}(a) ≠ 0 if n is odd then f(x) has neither local maximum nor local minimum at x = a and this is point of inflexion,

If n is even then f^{n}(a) < 0

f(x) has a local maximum at x = a and if f^{n}(a) > 0 then f(x) has a local minimum at x = a.

3. Working rule for finding Maxima & Minima

(i) Find the differential coefficient of f(x) with respect to x, i.e. f ‘(x) and equate it to zero.

(ii) Find different real values of x by solving the equation f ‘(x) = 0. Let its roots be a, b, c, ……..

(iii) Find the value of f”(x) and substitute the value of a_{1}, a_{2}, a_{3} ….in it and get the sign of f'(x) for each value of x.

(iv) If f'(a) < 0 then the value of f(x) is maximum at x = a and if f'(a) > 0 then value of f(x) will be minimum at x = a. Similarly by getting the signs of f”(x) at other points b, c we can find the points of maxima and minima.

4. Greatest & Least values of a function in a given interval

If a function f(x) is defined in an interval [a, b], then greatest or least values of this function occurs either at x = a or x = b or at those values of x where f'(x) = 0.

Thus greatest value of f(x) in interval [a, b] = max. [f (a), f(b), f(c)] Least value of f(x) in interval [a, b] = Min. [f(a), f(b), f(c)] where x = c is a point such that f'(c) = 0

5. How to find local maxima or Minima:

Step-I: Put f'(x) = 0 solve for values of x, let x = x_{1}, x_{2}, ……..

Step-II: Let x = x_{1} is now checking for local maxima or minima. Calculate f'(x_{1} – h) and f'(x_{1} + h)

Step-III: If h > 0 and h is very very small –

- f'(x
_{1}– h) < 0 and f'(x_{1}+ h) > 0 then x_{1}is point of minimum. - If f'(x
_{1}– h) > 0 and f'(x_{1}+ h) < 0 then x_{1}is point of maximum. - If f'(x
_{1}– h) & f'(x_{1}+ h) has same sign then x_{1}is neither point, of maximum nor point of minimum.

In the same way other values of x = x_{2}, x_{3} ….are checked, separately.

6. Properties of Maxima & Minima

- Between two equal values of f(x), there lie at least one maxima or minima.
- Maxima and minima occur alternately.
- When x passes a maximum point, the sign of f'(x) changes from +ve to -ve, whereas x passes through a minimum point, the sign of f'(x) changes from – ve to +ve.
- If there is no change in the sign of dy/dx on two sides of a point, then such a point is not an extreme point.
- If f(x) is a maximum (minimum) at a point x = a, then l/f(x), [f(x) ≠ 0] will be minimum (maximum) at that point.

7. Some standard geometrical results related to Maxima & Minima

- The area of rectangle with given perimeter is greatest when it is a square.
- The perimeter of a rectangle with given area is least when it is a square.
- The greatest rectangle inscribed in a given circle is a square.
- The greatest triangle inscribed in a given circle is equilateral.
- The semi vertical angle of a cone with given slant height and maximum volume is tan
^{-1}\(\sqrt{2}\). - The height of a cylinder of maximum volume inscribed in a sphere of radius a is 2a /\(\sqrt{3}\).