6.1 Extreme Values and Critical Points

Extreme values and critical points are key concepts in calculus, helping us find the highest and lowest points of functions. These tools are crucial for optimization problems, allowing us to determine maximum and minimum values in real-world scenarios.

By identifying critical points where a function's derivative is zero or undefined, we can pinpoint potential extrema. This knowledge forms the foundation for more advanced applications in curve sketching and problem-solving throughout calculus.

Extrema Types

Absolute Extrema

Top images from around the web for Absolute Extrema

• 

  Use a graph to locate the absolute maximum and absolute minimum | College Algebra View original

  Is this image relevant?

• 

  Maxima and Minima · Calculus View original

  Is this image relevant?

• 

  Use a graph to locate the absolute maximum and absolute minimum | MATH 1314: College Algebra View original

  Is this image relevant?

• 

  Use a graph to locate the absolute maximum and absolute minimum | College Algebra View original

  Is this image relevant?

• 

  Maxima and Minima · Calculus View original

  Is this image relevant?

1 of 3

Top images from around the web for Absolute Extrema

• 

  Use a graph to locate the absolute maximum and absolute minimum | College Algebra View original

  Is this image relevant?

• 

  Maxima and Minima · Calculus View original

  Is this image relevant?

• 

  Use a graph to locate the absolute maximum and absolute minimum | MATH 1314: College Algebra View original

  Is this image relevant?

• 

  Use a graph to locate the absolute maximum and absolute minimum | College Algebra View original

  Is this image relevant?

• 

  Maxima and Minima · Calculus View original

  Is this image relevant?

1 of 3

• Absolute maximum represents the highest value of a function over its entire domain
  • Can be determined by comparing all local maxima and the function values at the endpoints of the domain (if the domain is closed and bounded)
  • Example: For the function $f(x) = -x^2 + 4x$ on the interval $[0, 4]$, the absolute maximum occurs at $x = 2$ with a value of $f(2) = 4$
• Absolute minimum represents the lowest value of a function over its entire domain
  • Can be determined by comparing all local minima and the function values at the endpoints of the domain (if the domain is closed and bounded)
  • Example: For the function $f(x) = x^2 - 4x + 5$ on the interval $[-1, 3]$, the absolute minimum occurs at $x = -1$ with a value of $f(-1) = 10$

Local Extrema

• Local maximum is a point where the function value is greater than or equal to the function values in its immediate vicinity
  • Occurs when the function changes from increasing to decreasing
  • Example: For the function $f(x) = x^3 - 3x^2 - 9x + 10$, a local maximum occurs at $x = -1$
• Local minimum is a point where the function value is less than or equal to the function values in its immediate vicinity
  • Occurs when the function changes from decreasing to increasing
  • Example: For the function $f(x) = x^3 - 3x^2 - 9x + 10$, a local minimum occurs at $x = 3$

Finding Critical Points

Critical Points and Fermat's Theorem

• Critical points are points where the derivative of a function is either zero or undefined
  • Can be used to identify potential local extrema and inflection points
  • To find critical points, set the first derivative equal to zero and solve for x, or identify points where the derivative is undefined
• Fermat's theorem states that if a function $f$ has a local extremum at a point $c$ and $f'(c)$ exists, then $f'(c) = 0$
  • Helps identify potential local extrema by finding points where the derivative is zero
  • Example: For the function $f(x) = x^3 - 3x^2 - 9x + 10$, setting $f'(x) = 3x^2 - 6x - 9 = 0$ yields critical points at $x = -1$ and $x = 3$

Closed Interval Method and Endpoint Extrema

• Closed interval method is used to find absolute extrema of a continuous function on a closed interval $[a, b]$
  • Steps: Find critical points in the interval, evaluate the function at the critical points and endpoints, and compare the values to determine the absolute maximum and minimum
• Endpoint extrema occur when the absolute maximum or minimum of a function on a closed interval is located at one of the endpoints of the interval
  • Must be considered along with critical points when using the closed interval method
  • Example: For the function $f(x) = x^2 - 4x + 5$ on the interval $[-1, 3]$, the absolute minimum occurs at the endpoint $x = -1$ with a value of $f(-1) = 10$