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Maxima and Minima

Maxima and minima are critical points in the study of differentiation, where a function reaches its highest or lowest values within a given interval. These points are essential for understanding the behavior of functions, optimization problems, and real-world applications.

Definitions

  • Local Maximum: A function ( f(x) ) has a local maximum at ( x = a ) if ( f(a) ) is greater than or equal to ( f(x) ) for all ( x ) in some open interval around ( a ). In other words, ( f(a) \geq f(x) ) for all ( x ) near ( a ).
  • Local Minimum: A function ( f(x) ) has a local minimum at ( x = b ) if ( f(b) ) is less than or equal to ( f(x) ) for all ( x ) in some open interval around ( b ). That is, ( f(b) \leq f(x) ) for all ( x ) near ( b ).
  • Global Maximum and Minimum: These are the highest and lowest values of ( f(x) ) on the entire domain.

Steps for Finding Maxima and Minima

  1. Find the First Derivative (( f'(x) )):

    • Differentiate the function to obtain ( f'(x) ), which gives the slope of the tangent line at any point.

  2. Set the First Derivative Equal to Zero:

    • Solve the equation ( f'(x) = 0 ) to find the critical points. These are potential points where the function could have a maximum or minimum.

  3. Determine the Nature of Each Critical Point:

    • Use either the Second Derivative Test or the First Derivative Test to classify the critical points.

Second Derivative Test

  • Find the second derivative, ( f''(x) ).
  • Evaluate ( f''(x) ) at each critical point:
    • If ( f''(x) > 0 ), the function is concave up at that point, indicating a local minimum.
    • If ( f''(x) < 0 ), the function is concave down at that point, indicating a local maximum.
    • If ( f''(x) = 0 ), the test is inconclusive, and you should use the first derivative test.

First Derivative Test

  • Examine the sign of ( f'(x) ) around each critical point:
    • If ( f'(x) ) changes from positive to negative, the critical point is a local maximum.
    • If ( f'(x) ) changes from negative to positive, the critical point is a local minimum.
    • If ( f'(x) ) does not change sign, the point is neither a maximum nor a minimum.

Example 1: Find the Maxima and Minima of ( f(x) = x^3 - 3x^2 + 4 )

  1. Find the First Derivative: [ f'(x) = 3x^2 - 6x ]

  2. Set ( f'(x) = 0 ) to Find Critical Points: [ 3x^2 - 6x = 0 \implies 3x(x - 2) = 0 ] Thus, ( x = 0 ) and ( x = 2 ) are critical points.

  3. Apply the Second Derivative Test:

    • Find the second derivative: [ f''(x) = 6x - 6 ]
    • Evaluate ( f''(x) ) at ( x = 0 ) and ( x = 2 ):
      • At ( x = 0 ), ( f''(0) = 6(0) - 6 = -6 ) (concave down, local maximum).
      • At ( x = 2 ), ( f''(2) = 6(2) - 6 = 6 ) (concave up, local minimum).

Example 2: Find the Maxima and Minima of ( f(x) = e^x \sin(x) )

  1. Find the First Derivative: [ f'(x) = e^x \sin(x) + e^x \cos(x) ]

  2. Set ( f'(x) = 0 ) to Find Critical Points: [ e^x (\sin(x) + \cos(x)) = 0 ] Since ( e^x \neq 0 ), solve ( \sin(x) + \cos(x) = 0 ).

  3. Classify the Critical Points Using the First Derivative Test:

    • Analyze the sign changes of ( f'(x) ) around the critical points to determine if they correspond to maxima, minima, or neither.

Practical Applications

  • Business and Economics: Finding the maximum profit or minimum cost.
  • Physics: Determining the highest or lowest points of a motion trajectory.
  • Engineering: Optimizing design parameters for maximum efficiency.

Maxima and minima provide critical insights into the behavior of functions, especially in optimization problems where finding the highest or lowest values is essential.

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