How to find maximum and minimum value of the function with example.

 To find the maximum and minimum values of a function, you need to calculate the derivative of the function and find the points where the derivative is equal to zero or undefined. These points are called critical points, and they can be either maximum, minimum, or inflection points.

Once you have found the critical points, you can use the second derivative test to determine whether each critical point is a maximum, minimum, or inflection point.

Here's an example:

Suppose you have the function \[ f(x) = x^3 - 6x^2 + 9x + 2.\]

1. First, we need to find the derivative of the function: \[f'(x) = 3x^2 - 12x + 9.\]

2. Next, we need to find the critical points by setting the derivative equal to zero: \[3x^2 - 12x + 9 = 0.\]

We can solve this equation by factoring or using the quadratic formula: \[(x - 1)(3x - 9) = 0, \] which gives us \[x = 1 \ and \ x = 3. \]

These are the critical points of the function.

3. Now, we need to use the second derivative test to determine whether each critical point is a maximum, minimum, or inflection point. \[f''(x) = 6x - 12.\]

\[At \ x = 1, \ we \ have \ f''(1) = -6, \] \[ which\ means \ that \ f(x) \ has \ a \ relative \ maximum \ at \ x = 1.\]

\[At \ x = 3, \ we \ have \ f''(3) = 6, \] \[which \ means \ that \ f(x) \ has \ a \ relative \ minimum \ at \ x = 3.\]

Therefore, the maximum value of the function is f(1) = 6 and the minimum value is f(3) = -4.

In summary, to find the maximum and minimum values of a function, we need to find the critical points by setting the derivative equal to zero, and then use the second derivative test to determine whether each critical point is a maximum, minimum, or inflection point.

Some workout Exercises:-

1. Find maximum value of the function \[f(x)= x^3+3x^2+5\] at x = 5.

2. Find minimum value of of a function \[f(x)= x^2+3x-7\] at x = 7.