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N.K. Sharma
N.K. Sharma
Asked: March 14, 20242024-03-14T08:58:57+05:30 2024-03-14T08:58:57+05:30In: B.Com

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.

By maxima or minima of a function, what do you mean? Explain what a function’s absolute minimum means. Describe the procedures for locating a function’s maximum and minimum.

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    1. Abstract Classes Power Elite Author
      2024-03-14T08:59:33+05:30Added an answer on March 14, 2024 at 8:59 am

      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.
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