How do you determine where the given function f(x)=(x+3)23−6 is concave up and where it is concave down?

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riasc31lj · Accepted Answer

Step 1In order to investigate concavity, we&#039;ll look at the sign of the second derivativef(x)=(x+3)23-6f′(x)=23(x+3)−13Notice that&amp;nbsp;x=−3&amp;nbsp;there is a cusp at which the tangent becomes vertical. (The derivative goes to&amp;nbsp;∞)f&#039;(x)=−29(x+3)−43&amp;nbsp;=-29(x+3)43=-29(x+334)The only place where f&#039;&#039; might change sign is at&amp;nbsp;x=−3But clearly the numerator is always negative, and the denominator, being a positive times a 4th power, is always positive.So f&#039;&#039; is always negative where it is defined.The graph is concave down on&amp;nbsp;−∞,&amp;nbsp;−3)&amp;nbsp;and on&amp;nbsp;(−3,&amp;nbsp;∞)Because of the cusp at&amp;nbsp;x=0, we cannot combine these two intervals.Here&#039;s the graph of f

How do you determine where the given function

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