Overview(Maxima and minima)- A function f(x) is said to be maximum at x = a if f(a) is greater than every other value of f(x) in the immediate neighborhood of x = a (i.e., f(x) ceases to increase but begins to increase at x = a. Similarly the minimum value of f(x) will be that value at x = b which is less than other values in the immediate neighborhood of x = b.

## **Maxima and minima of function of two variables**

As we know that the value of a function at maximum point is called maximum value of a function. Similarly the value of a function at minimum point is called minimum value of a function.

The maxima and minima of a function is an extreme biggest and extreme smallest point of a function in a given range (interval) or entire region. Pierre de Fermat was the first mathematician to discover a general method for calculating maxima and minima of a function. The maxima and minima complement each other.

**Maxima and Minima of a function of one variables**

If f(x) is a single valued function defined in a region R then

Maxima is a maximum point if and only if

Minima is a minimum point if and only if .

**Maxima and Minima of a function of two independent variables**

Let f(x, y) be a defined function of two independent variables.

Then the point x = a and y = b is said to be a maximum point of f(x, y) if

For all positive and negative values of h and k.

Similarly the point x = a and y = b is said to be a minimum point of f(x, y) if

For all positive and negative values of h and k.

**Saddle point**

Critical points of a function of two variables are those points at which both partial derivatives of the function are zero. A critical point of a function of a single variable is either a local maximum, a local minimum, or neither. With functions of two variables there is a fourth possibility – a saddle point.

**Note**

1. Maximum and minimum values of a function occur alternatively

2. Function may have several maximum and minimum values in an interval

3. At some point the maximum value may be less than the minimum value

4. The points at which a function has maximum or minimum value are called turning points and the maximum and minimum values are known as extreme values, or extremum or turning values.

5. The values of x for which f(x) = 0 are often called critical values

**Criteria for maximum and minimum**

For a function y = f(x) to attain a maximum point at x = a,

For minimum point-

**Conditions for max. and min-**

**Necessary Condition**– If a function f(x) is maximum or minimum at a point x = b and if f’(b) exists then f’ (b) = 0.

**Sufficient Condition-** If b is a point in an interval where f(x) is defined and if f ‘(b) = 0 and f’’(b) is not equal to 0, then f(b) is maximum if f’’(b) <0 and is minimum if f’’(b) > 0. (The proof is not shown at present).

**Working Rule:**

**First derivative method**

To find the maximum or minimum point of a curve y = f(x).

Find f ‘ (x) and equate it to zero. From the equation f ‘(x) = 0, find the value of x, say a and b.

Here the number of roots of f ‘(x) = 0 will be equal to the number of degree of f ‘(x) = 0.

Then find f ‘(a – h) and f ‘(a + h), then note the change of sign if any (here h is very small).

If the change is from positive to negative, f(x) will be maximum at x = a. If again the change of sign is from negative to positive, f(x) will be maximum at x = a.

Similarly for x = b.

**Second derivative method-**

First we find the first derivative of y = f(x) i.e dy/dx and make it zero.

From the equation dy/dx = 0 find the value of x say a and b.

The again we find the second derivative of y or

Put x = a in , if at x = a is negative then the function is maximum at x = a and maximum value will be f(a).

If the value of at x = a is positive, then the function is minimum and the minimum value will be f(a)

Similarly we take for x = b.

**Example: Examine for maximum and minimum for the function **

Sol.

Here the first derivative is-

So that, we get-

Now we will get to know that the function is maximum or minimum at these values of x.

For x = 3

Let us assign to x, the values of 3 – h and 3 + h (here h is very small) and put these values at f(x).

Then-

which is negative for h is very small

which is positive.

Thus f’(x) changes sign from negative to positive as it passes through x = 3.

So that f(x) is minimum at x = 3 and the minimum value is-

And f(x) is maximum at x = -3.

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