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Critical Points

Definition (Critical Point) A critical point of a function defined on an open set critical points _gr_1.gif] is a point critical points _gr_2.gif] in critical points _gr_3.gif] where either one of the following is true: (i) critical points _gr_4.gif] or (ii) at least one of critical points _gr_5.gif] or critical points _gr_6.gif] does not exist at critical points _gr_7.gif] A critical point critical points _gr_8.gif] is called a saddle point of critical points _gr_9.gif] if every open disk centered at critical points _gr_10.gif] contains points in the domain of critical points _gr_11.gif] that satisfy critical points _gr_12.gif] as well as points in the domain of critical points _gr_13.gif] that satisfy critical points _gr_14.gif]

Example (Critical Point) Find the critical points for the given functions.

(a) Let critical points _gr_15.gif] Then

critical points _gr_16.gif]

These partial derivatives are equal to 0 when critical points _gr_17.gif] and critical points _gr_18.gif] so the only critical point is critical points _gr_19.gif] By completing the square we find that

critical points _gr_20.gif]

Since critical points _gr_21.gif] and critical points _gr_22.gif] we have critical points _gr_23.gif] for all values of critical points _gr_24.gif] and critical points _gr_25.gif] Therefore, critical points _gr_26.gif] is a local minimum, and in fact it is the absolute minimum of critical points _gr_27.gif] This can be confirmed geometrically from the graph of critical points _gr_28.gif], which is the elliptic paraboloid with vertex critical points _gr_29.gif] as shown.

critical points _gr_30.gif]

(b) Find the extreme values of critical points _gr_31.gif]

    Solution. Since critical points _gr_32.gif] and critical points _gr_33.gif] the only critical point is critical points _gr_34.gif] Notice that for points on the critical points _gr_35.gif]-axis we have critical points _gr_36.gif] so critical points _gr_37.gif] (if critical points _gr_38.gif]) However for points on the critical points _gr_39.gif]-axis we have critical points _gr_40.gif] so critical points _gr_41.gif] (if critical points _gr_42.gif]) Thus every disk with center critical points _gr_43.gif] contains points where critical points _gr_44.gif] takes positive values as well as points where critical points _gr_45.gif] takes negative values. Therefore critical points _gr_46.gif] cannot be an extreme value for critical points _gr_47.gif] so critical points _gr_48.gif] has no extreme values. This example illustrates the fact that a function need not have a maximum or minimum value at a critical point. The graph of critical points _gr_49.gif] is the hyperbolic paraboloid which has a horizontal tangent plane critical points _gr_50.gif] at the origin. You can see that critical points _gr_51.gif] is a maximum in the direction of the critical points _gr_52.gif] but not in the direction of the critical points _gr_53.gif]-axis. Near the origin the graph has the shape of a saddle.

critical points _gr_54.gif]

critical points _gr_55.gif]

critical points _gr_56.gif] Recommended Reading
critical points _gr_57.gif]functions of several variables
critical points _gr_58.gif]graphs of functions
critical points _gr_59.gif]polynomial functions
critical points _gr_60.gif]rational functions
critical points _gr_61.gif]level curves
critical points _gr_62.gif]level surfaces
critical points _gr_63.gif]limits of multivariate functions
critical points _gr_64.gif]continuity of multivariate functions
critical points _gr_65.gif]partial derivatives
critical points _gr_66.gif]higher order partial derivatives
critical points _gr_67.gif]tangent planes
critical points _gr_68.gif]total differential
critical points _gr_69.gif]linear approximation with multivariate functions
critical points _gr_70.gif]differentiability
critical points _gr_71.gif]chain rule with one independent parameter
critical points _gr_72.gif]chain rule with two independent parameters
critical points _gr_73.gif]chain rule with several independent parameters
critical points _gr_74.gif]directional derivatives
critical points _gr_75.gif]the gradient
critical points _gr_76.gif]the gradient and directional derivatives
critical points _gr_77.gif]steepest ascent and steepest descent
critical points _gr_78.gif]normal property of the gradient
critical points _gr_79.gif]tangent planes and normal lines
critical points _gr_80.gif]relative extrema
critical points _gr_81.gif]critical points
critical points _gr_82.gif]second partials test
critical points _gr_83.gif]absolute extrema
critical points _gr_84.gif]lagrange multipliers with one parameter
critical points _gr_85.gif]lagrange multipliers with two parameters

critical points _gr_86.gif] Recommended Math Books
critical points _gr_87.gif] Thomas' Calculus, Early Transcendentals, Media Upgrade (11th Edition)
 critical points _gr_88.gif] Thomas' Calculus, Media Upgrade (11th Edition)
 critical points _gr_89.gif] Thomas' Calculus Early Transcendentals; Student's Solutions Manual; Part One
critical points _gr_90.gif] Calculus (With Analytic Geometry)(8th edition)
critical points _gr_91.gif] Calculus (Stewart's Calculus Series)
 critical points _gr_92.gif] Applied Calculus
critical points _gr_93.gif] Calculus Textbooks
critical points _gr_94.gif] Elementary Calculus
critical points _gr_95.gif] Advanced Calculus
critical points _gr_96.gif] Supplementary Resources

critical points _gr_97.gif] Recommended Math Gifts
 critical points _gr_98.gif] Math Happy
critical points _gr_99.gif] Calculus Happy
critical points _gr_100.gif] Homework Happy
critical points _gr_101.gif] Limits Happy
critical points _gr_102.gif] I Love Math
critical points _gr_103.gif] I Love Calculus
critical points _gr_104.gif] I Love Homework
critical points _gr_105.gif] I Love Multivariate Calculus
critical points _gr_106.gif] I Love Limits

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Critical Points
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/critical-points.html
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