How do you determine whether the function&nbsp;f(x)=x8(ln(x))&nbsp;is concave up or concave down and its intervals?

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How do you determine whether the function&amp;nbsp;f(x)=x8(ln(x))&amp;nbsp;is concave up or concave down and its intervals?

muthe2ulj · Accepted Answer

Step 1The second derivative determines the concavity intervals. Inflection points occur when the sign of the second derivative changes at the locations where it equals zero.Df=R+f′(x)=8x7ln⁡x+x81x=8x7ln⁡x+x7=x7(8ln⁡x+1)fx)=7x6(8ln⁡x+1)+x78x=7x6(8ln⁡x+1)+8x6fx)=x6(56ln⁡x+15)fx)=0⇔x6(56ln⁡x+15)=0⇔(x6=0∨56ln⁡x+15=0)x6=0⇔x=0∈DfStep 256lnx+15=0⇔56lnx=-15⇔lnx=-1556⇔x=e-1556⇔x=1e1556≈0.765&amp;gt;0∈Df∀x∈(0,1e1556:f&quot;(x)&amp;lt;0&amp;nbsp;function is concave down∀x∈(1e1556,∞):f&quot;(x)&amp;gt;0&amp;nbsp;function is concave upNote: the sign of f&quot;(x) depends only on&amp;nbsp;56ln⁡x+15&amp;nbsp;since&amp;nbsp;x6&amp;nbsp;is always greater then zero for&amp;nbsp;x&amp;gt;0.

How do you determine whether the function PSKf(x)=x^{8}(\ln(x))

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