Derivatives - Maximum & Minimum Value of a Function

• Introduction to maximum and minimum values of a function

• Definition of local maximum and local minimum

• Relationship between derivatives and maximum/minimum values

• Steps to find maximum and minimum values of a function

• Example: Find the maximum and minimum values of a quadratic function

• Local maximum and local minimum in graphical representation

• Critical points of a function

• The first derivative test for identifying maximum and minimum points

• Example: Determine the intervals in which the function is increasing or decreasing

• The second derivative test for determining maximum and minimum points

• The concavity of a function

• Points of inflection

• Example: Determine the points of inflection of a cubic function

• Absolute maximum and absolute minimum

• Finding absolute maximum and minimum in closed intervals

• Example: Find the absolute maximum and minimum values of a trigonometric function

• Summary of finding maximum and minimum values of a function

• Important concepts and formulas

• Practice problems to reinforce understanding

• Conclusion and key takeaways

• Further resources and references for additional study

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Derivatives - Maximum & Minimum Value of a Function

Slide 11:

• Recap of the first derivative test for identifying maximum and minimum points:
  • If the derivative is positive to the left of a critical point and negative to the right, it indicates a local minimum.
  • If the derivative is negative to the left of a critical point and positive to the right, it indicates a local maximum.
  • If the derivative does not change sign at a critical point, further tests are required.

Slide 12:

• The second derivative test for determining maximum and minimum points:
  • If the second derivative is positive at a critical point, it indicates concavity upwards and a local minimum.
  • If the second derivative is negative at a critical point, it indicates concavity downwards and a local maximum.
  • If the second derivative is zero at a critical point, the test is inconclusive.

Slide 13:

• Importance of concavity of a function:
  • A function is concave upwards if its second derivative is positive.
  • A function is concave downwards if its second derivative is negative.
  • Points where the concavity changes are called points of inflection.

Slide 14:

• Finding points of inflection of a function:
  • Locate the critical points by finding where the second derivative is zero or undefined.
  • Test the concavity of the function at these critical points.
  • If the concavity changes from upwards to downwards or vice versa, it indicates a point of inflection.

Slide 15:

• Definition of absolute maximum and absolute minimum:
  • The absolute maximum of a function is the highest value it reaches over its entire domain.
  • The absolute minimum of a function is the lowest value it reaches over its entire domain.

Slide 16:

• Finding absolute maximum and minimum in closed intervals:
  • Evaluate the function at its critical points and endpoints of the closed interval.
  • The largest value obtained will be the absolute maximum and the smallest value will be the absolute minimum.

Slide 17:

• Example:
  • Find the absolute maximum and minimum values of the function f(x) = sin(x) in the closed interval [0, π/2].
  • Evaluate f(x) at x = 0, x = π/2, and critical points (if any).
  • Determine the largest and smallest values obtained.

Slide 18:

• Summary of finding maximum and minimum values of a function:
  • Use the first derivative test to identify local maximum and minimum points.
  • Use the second derivative test to confirm the nature of these points.
  • Determine points of inflection where the concavity changes.
  • Find absolute maximum and minimum values in closed intervals.

Slide 19:

• Important concepts and formulas:
  • First derivative test: f’(x) > 0 indicates increasing; f’(x) < 0 indicates decreasing.
  • Second derivative test: f’’(x) > 0 indicates concave upwards; f’’(x) < 0 indicates concave downwards.
  • Points of inflection: Locations where concavity changes.
  • Absolute maximum and minimum: Highest and lowest values achieved by a function over its entire domain.

Slide 20:

• Practice problems:
  1. Find the maximum and minimum values of the function f(x) = 3x^2 - 6x + 2.
  2. Determine the points of inflection of the function g(x) = x^3 - 3x^2 + 2x.
  3. Find the absolute maximum and minimum values of the function h(x) = cos(x) in the closed interval [-π, π]. ``

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Derivatives - Maximum & Minimum Value of a Function

Slide 11:

• Recap of the first derivative test for identifying maximum and minimum points:
  • If the derivative is positive to the left of a critical point and negative to the right, it indicates a local minimum.
  • If the derivative is negative to the left of a critical point and positive to the right, it indicates a local maximum.
  • If the derivative does not change sign at a critical point, further tests are required.

Slide 12:

• The second derivative test for determining maximum and minimum points:
  • If the second derivative is positive at a critical point, it indicates concavity upwards and a local minimum.
  • If the second derivative is negative at a critical point, it indicates concavity downwards and a local maximum.
  • If the second derivative is zero at a critical point, the test is inconclusive.

Slide 13:

• Importance of concavity of a function:
  • A function is concave upwards if its second derivative is positive.
  • A function is concave downwards if its second derivative is negative.
  • Points where the concavity changes are called points of inflection.

Slide 14:

• Finding points of inflection of a function:
  • Locate the critical points by finding where the second derivative is zero or undefined.
  • Test the concavity of the function at these critical points.
  • If the concavity changes from upwards to downwards or vice versa, it indicates a point of inflection.

Slide 15:

• Definition of absolute maximum and absolute minimum:
  • The absolute maximum of a function is the highest value it reaches over its entire domain.
  • The absolute minimum of a function is the lowest value it reaches over its entire domain.

Slide 16:

• Finding absolute maximum and minimum in closed intervals:
  • Evaluate the function at its critical points and endpoints of the closed interval.
  • The largest value obtained will be the absolute maximum and the smallest value will be the absolute minimum.

Slide 17:

• Example:
  • Find the absolute maximum and minimum values of the function f(x) = sin(x) in the closed interval [0, π/2].
  • Evaluate f(x) at x = 0, x = π/2, and critical points (if any).
  • Determine the largest and smallest values obtained.

Slide 18:

• Summary of finding maximum and minimum values of a function:
  • Use the first derivative test to identify local maximum and minimum points.
  • Use the second derivative test to confirm the nature of these points.
  • Determine points of inflection where the concavity changes.
  • Find absolute maximum and minimum values in closed intervals.

Slide 19:

• Important concepts and formulas:
  • First derivative test: f’(x) > 0 indicates increasing; f’(x) < 0 indicates decreasing.
  • Second derivative test: f’’(x) > 0 indicates concave upwards; f’’(x) < 0 indicates concave downwards.
  • Points of inflection: Locations where concavity changes.
  • Absolute maximum and minimum: Highest and lowest values achieved by a function over its entire domain.

Slide 20:

• Practice problems:
  1. Find the maximum and minimum values of the function f(x) = 3x^2 - 6x + 2.
  2. Determine the points of inflection of the function g(x) = x^3 - 3x^2 + 2x.
  3. Find the absolute maximum and minimum values of the function h(x) = cos(x) in the closed interval [-π, π].

Derivatives - Maximum & Minimum Value of a Function

Slide 21:

• Recap of the example:
  • Find the maximum and minimum values of the function f(x) = 3x^2 - 6x + 2.
  • Differentiate the function to find its derivative: f’(x) = 6x - 6.
  • Set the derivative equal to zero and solve for x: 6x - 6 = 0.
  • Find the critical point: x = 1.
  • Test the function to the left and right of the critical point.
  • Determine that it is a minimum point.

Slide 22:

• Example continued:
  • Differentiate again to find the second derivative: f’’(x) = 6.
  • Evaluate the second derivative at the critical point: f’’(1) = 6.
  • Since the second derivative is positive, it confirms that the point is a local minimum.

Slide 23:

• Recap of the example:
  • Determine the points of inflection of the function g(x) = x^3 - 3x^2 + 2x.
  • Differentiate the function to find its derivative: g’(x) = 3x^2 - 6x + 2.
  • Set the derivative equal to zero and solve for x: 3x^2 - 6x + 2 = 0.
  • Find the critical points: x = 1 and x = 2.
  • Evaluate the second derivative to determine the concavity of the function.

Slide 24:

• Example continued:
  • Differentiate again to find the second derivative: g’’(x) = 6x - 6.
  • Evaluate the second derivative at the critical points: g’’(1) = 0 and g’’(2) = 6.
  • The second derivative test is inconclusive as the second derivative does not change sign at the critical points.

Slide 25:

• Recapping the concavity test:
  • If the second derivative is positive, the function is concave upwards.
  • If the second derivative is negative, the function is concave downwards.
  • If the second derivative is zero, further tests are required.

Slide 26:

• Recap of the example:
  • Find the absolute maximum and minimum values of the function h(x) = cos(x) in the closed interval [-π, π].
  • Evaluate the function at the critical points and endpoints:
    • h(-π) = -1, h(π) = -1, h(0) = 1.
  • The absolute maximum value is 1 and the absolute minimum value is -1.

Slide 27:

• Summary of the steps to find maximum and minimum values of a function:
  1. Find the derivative of the function.
  2. Set the derivative equal to zero to find critical points.
  3. Test the derivative to determine the nature of the critical points (using the first derivative test).
  4. Differentiate again to find the second derivative.
  5. Evaluate the second derivative at the critical points (using the second derivative test).
  6. Determine the concavity of the function and locate points of inflection.
  7. Evaluate the function at critical points and endpoints to find the absolute maximum and minimum values.

Slide 28:

• Key takeaways from today’s lesson:
  • Derivatives help determine maximum and minimum values of a function.
  • Critical points and concavity are crucial in identifying these extreme values.
  • The second derivative test confirms the nature of the critical points.
  • Points of inflection indicate where the concavity changes.
  • Absolute maximum and minimum values are found in closed intervals.

Slide 29:

• Additional resources for further study:
  • “Calculus” by James Stewart
  • “Introduction to Calculus and Analysis” by Richard Courant and Fritz John
  • Online resources such as Khan Academy and MIT OpenCourseWare

Slide 30:

• Thank you for attending today’s lecture!
  • Stay tuned for the next topic: Applications of Derivatives.
  • Don’t forget to practice and ask questions if you need further clarification.
  • Have a great day!

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