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Formula
Calculate the differentiation
Find the points of local maxima, local minima and the points of inflection of the function
Graph
$f \left( x \right) = \left ( x ^ { 2 } - 2 \right ) ^ { 2 }$
$x$Intercept
$\left ( - \sqrt{ 2 } , 0 \right )$, $\left ( \sqrt{ 2 } , 0 \right )$
$f \left( x \right)$Intercept
$\left ( 0 , 4 \right )$
Derivative
$4 x ^ { 3 } - 8 x$
Seconde derivative
$12 x ^ { 2 } - 8$
Local Minimum
$\left ( - \sqrt{ 2 } , 0 \right )$, $\left ( \sqrt{ 2 } , 0 \right )$
Local Maximum
$\left ( 0 , 4 \right )$
Point of inflection
$\left ( - \dfrac { \sqrt{ 6 } } { 3 } , \dfrac { 16 } { 9 } \right )$, $\left ( \dfrac { \sqrt{ 6 } } { 3 } , \dfrac { 16 } { 9 } \right )$
$f \left( x \right) = \left( x ^{ 2 } -2 \right) ^{ 2 }$
$\dfrac {d } {d x } {\left( f \left( x \right) \right)} = 4 x ^ { 3 } - 8 x$
Calculate the differentiation of the logarithmic function
$\dfrac {d } {d \color{#FF6800}{ x } } {\left( \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \right ) ^ { \color{#FF6800}{ 2 } } \right)}$
 Calculate the differentiation 
$\color{#FF6800}{ 4 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 3 } } \color{#FF6800}{ - } \color{#FF6800}{ 8 } \color{#FF6800}{ x }$
$x = - \sqrt{ 2 } ,$ minimal value $\\ x = 0 ,$ maximal value $\\ x = \sqrt{ 2 } ,$ minimal value 
Find the points of local maxima, local minima and the points of inflection of the function
$f \left( \color{#FF6800}{ x } \right) = \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \right ) ^ { \color{#FF6800}{ 2 } }$
 Find critical points (Points where the differential value becomes 0) 
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 2 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ 0 } \\ \color{#FF6800}{ x } = \sqrt{ \color{#FF6800}{ 2 } } \end{array}$
$\begin{cases} f \left( \color{#FF6800}{ x } \right) = \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \right ) ^ { \color{#FF6800}{ 2 } } \\ \dfrac {d } {d \color{#FF6800}{ x } } {\left( f \right)} \left( \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 2 } } \right) = \color{#FF6800}{ 0 } \end{cases}$
 Determine if it is the maximal value, the minimal value, the increasing inflection point, or the decreasing inflection point 
 It is the minimal value 
$\begin{cases} f \left( \color{#FF6800}{ x } \right) = \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \right ) ^ { \color{#FF6800}{ 2 } } \\ \dfrac {d } {d \color{#FF6800}{ x } } {\left( f \right)} \left( \color{#FF6800}{ 0 } \right) = \color{#FF6800}{ 0 } \end{cases}$
 Determine if it is the maximal value, the minimal value, the increasing inflection point, or the decreasing inflection point 
 It is the maximal value 
$\begin{cases} f \left( \color{#FF6800}{ x } \right) = \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \right ) ^ { \color{#FF6800}{ 2 } } \\ \dfrac {d } {d \color{#FF6800}{ x } } {\left( f \right)} \left( \sqrt{ \color{#FF6800}{ 2 } } \right) = \color{#FF6800}{ 0 } \end{cases}$
 Determine if it is the maximal value, the minimal value, the increasing inflection point, or the decreasing inflection point 
 It is the minimal value 
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Higher order function
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