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Extrema and Critical Points in Calculus

Learn how to find, classify, and interpret extrema and critical points in calculus, including maximums, minimums, and points of inflection. Discover how critical points impact functions and how to identify them on graphs.

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Extrema and Critical Points in Calculus

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  1. 3.6 Critical Points and Extrema Objective: Find the extrema of a function.

  2. Critical Points: Points at which the nature of a graph changes. (points at which a line drawn tangent to the curve is horizontal or vertical) 3 Types of Critical Points: -Maximums -Minimums -Points of Inflection Absolute Max: A point that represents the maximum value a function assumes over its domain.

  3. Absolute Min.: A point that represents the minimum value a function assumes over its domain. A max. or min. value of a function. A point that represents the max. or min. for a certain interval. Extremum: Relative Extrema:

  4. Ex. 1) Locate the extrema for the graph of y=f(x). Name and classify the extrema of the function. Ex. 2) Use a graphing calculator to graph f(x) = x³ - 8x + 3 and to determine and classify its extrema. See pg. 174 table.

  5. Ex. 3) The function f(x) = 3x^4 – 4x³ has critical points at x=0 and x=1. Determine whether each of these critical points is the location of a max., min., or a point of inflection. Ex. 4) One hour after x milligrams of particular drug are given to person, the rise in body temp., T(x), in degrees Fahrenheit is given by T(x)= x – (x²/9). The model has a critical point at x= 4.5. a.)Determine if this critical point is a max. b.) Why should a doctor be aware of this critical point?

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