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the graph of y=(x^2-4)^4(x^2+1)^ 5 is shown to the right. Find the coordinates of the local maximum points and minimum points

the graph of y=(x^2-4)^4(x^2+1)^ 5 is shown to the right. Find the coordinates of-example-1

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Step 1

Given;


y=(x^2-4)^4(x^2+1)^5

Required; To find the coordinates of the local minima and maxima

Step 2

Find the local minima and maxima


\begin{gathered} \mathrm{Suppose\:that\:}x=c\mathrm{\:is\:a\:critical\:point\:of\:}f\left(x\right)\mathrm{\:then,\:} \\ \mathrm{If\:}f\:'\left(x\right)>0\mathrm{\:to\:the\:left\:of\:}x=c\mathrm{\:and\:}f\:'\left(x\right)<0\mathrm{\:to\:the\:right\:of\:}x=c\mathrm{\:then\:}x=c\mathrm{\:is\:a\:local\:maximum.} \\ \mathrm{If\:}f\:'\left(x\right)<0\mathrm{\:to\:the\:left\:of\:}x=c\mathrm{\:and\:}f\:'\left(x\right)>\:0\mathrm{\:to\:the\:right\:of\:}x=c\mathrm{\:then\:}x=c\mathrm{\:is\:a\:local\:minimum.} \\ \mathrm{If\:}f\:'\left(x\right)\mathrm{\:is\:the\:same\:sign\:on\:both\:sides\:of\:}x=c\mathrm{\:then\:}x=c \\ \mathrm{\:is\:neither\:a\:local\:maximum\:nor\:a\:local\:minimum.} \end{gathered}

Therefore; f'(x) is given as;


f^(\prime)(x)=2x(x^2-4)^3(x^2+1)^4[9x^2-16]

Step 3

Find the increasing and decreasing intervals from the graph


\begin{gathered} Decreasing;-\inftyPlugin x=-2 into y[tex]\begin{gathered} \mathrm{Minimum}\left(-2,\:0\right) \\ \end{gathered}

Plugin -4/3 into y


\mathrm{Maximum}\left(-(4)/(3),\:(5^(14)\cdot \:256)/(387420489)\right)

Plugin x=0 into y


\mathrm{Minimum}\left(0,\:256\right)

Plugin x=4/3


\mathrm{Maximum}\left((4)/(3),\:(5^(14)\cdot \:256)/(387420489)\right)

Plugging x=2 into y


\mathrm{Minimum}\left(2,\:0\right)

Answer; The maximum points are;


\begin{gathered} ((4)/(3),(5^(14)*256)/(387420489))\text{ or \lparen1.33},4033.09) \\ \left(-(4)/(3),\:(5^(14)\cdot\:256)/(387420489)\right)or\text{ \lparen-1.33},\text{ 4033.09\rparen} \end{gathered}

The minimum points are ;


\begin{gathered} \left(-2,\:0\right) \\ \left(0,\:256\right) \\ \left(2,\:0\right) \end{gathered}

User Carlos Espinoza
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