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MAXIMA AND MINIMA

The function f(x) is said to have attained its maximum value for x=a, if the function ceases to increase and begins to decrease at x= a. A function f(x) is said to have attained its minimum value for x= b, if the function ceases to decrease and begins to increase at x= b.

In the figure we can see the maximum points and minimum points

MATHEMATICALLY

• A value f(x) is said to be the absolute maximum of a function f(x) in an interval [a,b] if  f(a) > = f(x) for every x E [a,b]

• A value f(b) is said to be the absolute minimum of a function f(x) in an interval [a,b] if f(b) < = f(x) for every x E [a,b].

• A value f(x₁) is said to be a local maximum of a function f(x) if f(x₁) is greater than any other value assumed by f(x) in the immediate neighbourhood of x = x₁. If f(x₁) is a local maximum, then f’ (x₁) = 0 and f”(x₁) < 0

• A value f(x₂) is said to be a local minimum of a function f(x) if f(x₂) is less than any other value assumed by f(x) in the immediate neighbourhood of x = x₂ .If f(x₂) is a local maximum, then f’(x₂) = 0 and f” (x₂) > 0

 To Understand the meaning of all the four definitions, refer to the figure given below.

x₁ and x₂are the points of local maximum and local minimum. We can see that if we take a small interval around x₁ and x₂ then these are the greatest and least values respectively in that small interval. Check also that f(x₁) = 0, f’(x₁) = 0. In this interval the absolute maximum is f(b) and f(a) is the absolute minimum.

CONDITIONS FOR MAXIMA AND MINIMA

1. At a maximum point, the function y=f(x) changes from an increasing to a decreasing state.

Therefore dy/dx changes from a positive value to a negative value

From this we can say, that, in changing from a positive value to a negative value dy/dx must pass through the value 0. Hence dy/dx = 0 at the maximum point.

2. At a minimum point, the function y=f(x) changes from a decreasing to an increasing state.

Therefore dy/dx changes from a negative value to a positive value

From this we can say, that, in changing from a negative value to a positive value dy/dx must pass through the value 0. Hence dy/dx = 0 at the minimum point also.

NOTE: It may happen that in spite of dy/dx being 0, the function may go on increasing or decreasing and it may not change from an increasing to a decreasing state or vice-versa. This point is then called the point of inflexion.

HOW TO CALCULATE THE MAXIMUM AND MINIMUM VALUES OF A FUNCTION f(x) IN ANY INTERVAL [a,b]

1. Find whether teh function f(x) is continuous in teh interval [a,b]

2. Then find the first derivative dy/dx of the function.

3. If f(x) is continuous on [a,b], then solve f’ (x) = 0. Suppose the solutions are x = x₁, x₂, x₃ . . .

4. Then find the second derivative of f(x), i.e. d²y/dx² or f”(x)

5. Substitute the values of x as obtained in 3 above. The values of x for which f”(x) is negative are the maximum values, and the values of x for which f”(x) is positive are the minimum values. 

NOTE: If in a particular equation d²y/dx² = 0 and even further if d³y/dx³= 0, then teh function f(x) is a maximum or minimum according as d³y/dx³=+ or -, or = d⁴y/dx⁴=+ or - respectively.