What do you mean by maxima or minima of a function? State the meaning of absolute minimum of a function. Explain the steps for finding maxima and minima of a function.

Maxima and Minima of a Function

1. Definition of Maxima and Minima:

• Maxima and minima refer to the highest and lowest points of a function, respectively.
• In mathematical terms, a function has a maximum at a point if the function value at that point is greater than or equal to the function values at all nearby points. Similarly, a function has a minimum at a point if the function value at that point is less than or equal to the function values at all nearby points.

2. Absolute Minimum of a Function:

• The absolute minimum of a function is the smallest value that the function takes on over its entire domain.
• It may occur at a single point or at multiple points.

3. Steps for Finding Maxima and Minima of a Function:

a) Find the derivative of the function:

  - The critical points of the function occur where the derivative is zero or undefined. 
  - Set the derivative equal to zero and solve for x to find the critical points.

b) Determine the nature of the critical points:

  - Use the second derivative test or the first derivative test to determine whether the critical points are maxima, minima, or points of inflection.
  - If the second derivative is positive at a critical point, it is a local minimum. If the second derivative is negative, it is a local maximum. If the second derivative is zero, the test is inconclusive.

c) Check endpoints and boundary points:

  - If the function is defined on a closed interval, check the function value at the endpoints and any other boundary points to determine if they are maxima or minima.

d) Determine the absolute minimum or maximum:

  - Compare the function values at the critical points, endpoints, and boundary points to find the absolute minimum or maximum of the function.

4. Example:

• Consider the function f(x) = x^2 – 4x + 3.
• Find the critical points by taking the derivative: f'(x) = 2x – 4.
• Set f'(x) = 0 to find the critical point: 2x – 4 = 0, x = 2.
• Check the nature of the critical point using the second derivative test: f''(x) = 2, which is positive, so the critical point x = 2 is a local minimum.

5. Conclusion:

• Finding the maxima and minima of a function involves identifying critical points, determining their nature, and comparing them to find the absolute minimum or maximum.
• This process is essential in optimization problems and is used extensively in calculus and mathematical modeling.