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4.6: Derivatives and the Shape of a Graph

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_Graph

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 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.

4.3: Derivatives and the Shape of a Graph

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_Graph

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 ...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 …

4.4: Derivatives and the Shape of a Graph

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_Graph

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 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.

4.4: Derivatives and the Shape of a Graph

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_Graph

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 ...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 …

4.4: Derivatives and the Shape of a Graph

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_Graph

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 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.

4.6: Derivatives and the Shape of a Graph

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_Graph

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 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.

4.5: Derivatives and the Shape of a Graph

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_Graph

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 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.

4.4: The Second Derivative Test and Curve Sketching

https://math.libretexts.org/Courses/College_of_Southern_Nevada/Calculus_(Hutchinson)/04%3A_Applications_of_Derivatives/4.04%3A_The_Second_Derivative_Test_and_Curve_Sketching

To determine concavity, we need to find the second derivative

$f''(x).$ The first derivative is

$f'(x)=3x^2−12x+9,$ so the second derivative is

$f''(x)=6x−12.$ If the function changes concavity,...To determine concavity, we need to find the second derivative

$f''(x).$ The first derivative is

$f'(x)=3x^2−12x+9,$ so the second derivative is

$f''(x)=6x−12.$ If the function changes concavity, it occurs either when

$f''(x)=0$ or

$f''(x)$ is undefined.

4.5: Derivatives and the Shape of a Graph

https://math.libretexts.org/Courses/De_Anza_College/Calculus_I%3A_Differential_Calculus/04%3A_Applications_of_Derivatives/4.05%3A_Derivatives_and_the_Shape_of_a_Graph

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 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.

4.3: Graphing Functions

https://math.libretexts.org/Courses/Chabot_College/MTH_1%3A_Calculus_I/04%3A_Applications_of_Derivatives/4.03%3A_Graphing_Functions

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 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.

3.3: Shapes of Curves

https://math.libretexts.org/Courses/Cosumnes_River_College/Math_400%3A_Calculus_I_-_Differential_Calculus/03%3A_Advancing_with_Applications/3.03%3A_Shapes_of_Curves

This section covers techniques for graphing functions by analyzing their shapes using calculus. It explains how the first and second derivatives indicate increasing/decreasing intervals, concavity, an...This section covers techniques for graphing functions by analyzing their shapes using calculus. It explains how the first and second derivatives indicate increasing/decreasing intervals, concavity, and inflection points. By applying these concepts, you can accurately sketch the function’s curve and identify key features such as peaks, valleys, and changes in direction.

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