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Analyze and sketch a graph of the function y = 6x^4 + 7x^3 . Find any intercepts, relative...

Question:

Analyze and sketch a graph of the function {eq}y = 6x^4 + 7x^3 {/eq}. Find any intercepts, relative extrema, points of inflection, and asymptotes.

Extrema Of A Function:

To Find the extrema of a function we use the following function

{eq}\mathrm{An\:inflection\:point\:is\:a\:point\:on\:the\:graph\:at\:which\:the\:second\:derivative\:changes\:sign.}\\ \mathrm{If\:}f\:''\left(x\right)>0\mathrm{\:then\:}f\left(x\right)\mathrm{\:concave\:upwards.}\\ \mathrm{If\:}f\:''\left(x\right)<0\mathrm{\:then\:}f\left(x\right)\mathrm{\:concave\:downwards.}\\ \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.}\\ {/eq}

Answer and Explanation:

{eq}\text{Consider the function}\\ y = 6x^4 + 7x^3 \\ {\color{Red}...

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Learn more about this topic:

Finding Local Extrema on a Graphing Calculator
Finding Local Extrema on a Graphing Calculator

from Saxon Calculus Homeschool: Online Textbook Help

Chapter 7 / Lesson 9
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