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- https://math.libretexts.org/Courses/City_University_of_New_York/Calculus_I_(CUNY)/04%3A_Applications_of_Derivatives/4.06%3A_Derivatives_and_the_Shape_of_a_GraphUsing the results from the previous section, we are now able to determine whether a critical point of a function actually corresponds to a local extreme value. In this section, we also see how the sec...Using the results from the previous section, we are now able to determine whether a critical point of a function actually corresponds to a local extreme value. In this section, we also see how the second derivative provides information about the shape of a graph by describing whether the graph of a function curves upward or curves downward.
- https://math.libretexts.org/Under_Construction/Purgatory/Remixer_University/Username%3A_hdagnew@ucdavis.edu/Courses%2F%2FRemixer_University%2F%2FUsername%3A_hdagnew@ucdavis.edu%2F%2FMonroe2/Courses%2F%2FRemixer_University%2F%2FUsername%3A_hdagnew@ucdavis.edu%2F%2FMonroe2%2F%2FChapter_4%3A_Applications_of_Derivatives/Courses%2F%2FRemixer_University%2F%2FUsername%3A_hdagnew@ucdavis.edu%2F%2FMonroe2%2F%2FChapter_4%3A_Applications_of_Derivatives%2F%2F4.3%3A_Derivatives_and_the_Shape_of_a_Graphlet f be a continuous function over an interval I containing a critical point c such that f is differentiable over I except possibly at c; if f′ changes sign from positive ...let f be a continuous function over an interval I containing a critical point c such that f is differentiable over I except possibly at c; if f′ changes sign from positive to negative as x increases through c, then f has a local maximum at c; if f′ changes sign from negative to positive as x increases through c, then f has a local minimum at c; if f′ does not change sign as x increases through c, then f does not have a …
- https://math.libretexts.org/Courses/Mission_College/Math_3A%3A_Calculus_I_(Reed)/04%3A_Applications_of_Derivatives/4.04%3A_Derivatives_and_the_Shape_of_a_GraphUsing the results from the previous section, we are now able to determine whether a critical point of a function actually corresponds to a local extreme value. In this section, we also see how the sec...Using the results from the previous section, we are now able to determine whether a critical point of a function actually corresponds to a local extreme value. In this section, we also see how the second derivative provides information about the shape of a graph by describing whether the graph of a function curves upward or curves downward.
- https://math.libretexts.org/Courses/Butler_Community_College/MA148%3A_Calculus_with_Applications_-_Butler_CC/04%3A_Functions_of_Two_Variables/4.03%3A_OptimizationThe partial derivatives tell us something about where a surface has local maxima and minima. Remember that even in the one-variable cases, there were critical points which were neither maxima nor mini...The partial derivatives tell us something about where a surface has local maxima and minima. Remember that even in the one-variable cases, there were critical points which were neither maxima nor minima – this is also true for functions of many variables. In fact, as you might expect, the situation is even more complicated.
- https://math.libretexts.org/Courses/Hope_College/Math_126_-_Calculus_with_Review_II/02%3A_Using_Derivatives/2.02%3A_Using_Derivatives_to_Identify_Extreme_ValuesThe critical numbers of a continuous function f are the values of p for which f′(p)=0 or f′(p) does not exist. These values are important because they identify horizontal tangent lines or corner p...The critical numbers of a continuous function f are the values of p for which f′(p)=0 or f′(p) does not exist. These values are important because they identify horizontal tangent lines or corner points on the graph, which are the only possible locations at which a local maximum or local minimum can occur.
- https://math.libretexts.org/Bookshelves/Calculus/Calculus_3e_(Apex)/12%3A_Functions_of_Several_Variables/12.08%3A_Extreme_ValuesGiven a function z=f(x,y) , we are often interested in points where z takes on the largest or smallest values.
- https://math.libretexts.org/Bookshelves/Calculus/Calculus_3e_(Apex)/03%3A_The_Graphical_Behavior_of_Functions/3.04%3A_Concavity_and_the_Second_DerivativeWe have been learning how the first and second derivatives of a function relate information about the graph of that function. We have found intervals of increasing and decreasing, intervals where the ...We have been learning how the first and second derivatives of a function relate information about the graph of that function. We have found intervals of increasing and decreasing, intervals where the graph is concave up and down, along with the locations of relative extrema and inflection points.
- https://math.libretexts.org/Courses/Borough_of_Manhattan_Community_College/MAT301_Calculus_I/04%3A_Applications_of_Derivatives/4.04%3A_Derivatives_and_the_Shape_of_a_Graphlet f be a continuous function over an interval I containing a critical point c such that f is differentiable over I except possibly at c; if f′ changes sign from positive ...let f be a continuous function over an interval I containing a critical point c such that f is differentiable over I except possibly at c; if f′ changes sign from positive to negative as x increases through c, then f has a local maximum at c; if f′ changes sign from negative to positive as x increases through c, then f has a local minimum at c; if f′ does not change sign as x increases through c, then f does not have a …
- https://math.libretexts.org/Courses/Penn_State_University_Greater_Allegheny/Math_140%3A_Calculus_1_(Gaydos)/04%3A_Applications_of_Derivatives/4.04%3A_Derivatives_and_the_Shape_of_a_GraphUsing the results from the previous section, we are now able to determine whether a critical point of a function actually corresponds to a local extreme value. In this section, we also see how the sec...Using the results from the previous section, we are now able to determine whether a critical point of a function actually corresponds to a local extreme value. In this section, we also see how the second derivative provides information about the shape of a graph by describing whether the graph of a function curves upward or curves downward.
- https://math.libretexts.org/Courses/Laney_College/Math_3A%3A_Calculus_1_(Fall_2022)/04%3A_Applications_of_Derivatives/4.06%3A_Derivatives_and_the_Shape_of_a_GraphUsing the results from the previous section, we are now able to determine whether a critical point of a function actually corresponds to a local extreme value. In this section, we also see how the sec...Using the results from the previous section, we are now able to determine whether a critical point of a function actually corresponds to a local extreme value. In this section, we also see how the second derivative provides information about the shape of a graph by describing whether the graph of a function curves upward or curves downward.
- https://math.libretexts.org/Courses/Coastline_College/Math_C180%3A_Calculus_I_(Tran)/04%3A_Applications_of_Derivatives/4.05%3A_Derivatives_and_the_Shape_of_a_GraphUsing the results from the previous section, we are now able to determine whether a critical point of a function actually corresponds to a local extreme value. In this section, we also see how the sec...Using the results from the previous section, we are now able to determine whether a critical point of a function actually corresponds to a local extreme value. In this section, we also see how the second derivative provides information about the shape of a graph by describing whether the graph of a function curves upward or curves downward.
