How do you determine whether the function f(x)=x2ex is concave up or concave down and its intervals?

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Kendall Clark · Accepted Answer

Step 1f&amp;nbsp;″(x)=0f(x)=x2×exf′(x)=2×x×ex+x2×exf&amp;nbsp;″(x)=2×ex+2×x×ex+2×x×ex+x2×ex==ex(2+4×x+x2)f&amp;nbsp;″(x)=0⇔2×ex(1+x+x2)=0Remember that&amp;nbsp;ex&amp;nbsp;is always positive⇒(1+x+x2)=0x1,2=−b±b2−4ac2ax1,&amp;nbsp;2=−4±42−4×2×12×1==−4±16−82==−4±82==−4±222==−2±2x1=−2+2=−0.59x2=−2−2=−3.41now you have 3 intervals:1.&amp;nbsp;(−∞,&amp;nbsp;x2)2.&amp;nbsp;(x2,&amp;nbsp;x1)3.&amp;nbsp;(x1,&amp;nbsp;∞)ask yourself if&amp;nbsp;f&amp;nbsp;″(x)&amp;nbsp;is positive or negative on each interval: (choose a number of the interval and calculate the value,don&#039;t choose&amp;nbsp;x1&amp;nbsp;or&amp;nbsp;x2&amp;nbsp;because there is 0 as we calculated before).1.&amp;nbsp;(−∞,&amp;nbsp;−3.41)x=4,&amp;nbsp;f&amp;nbsp;″(−4)&amp;gt;02.&amp;nbsp;(−3.41,&amp;nbsp;−0.59)x=−1,&amp;nbsp;f&amp;nbsp;″(−1)&amp;lt;03.&amp;nbsp;(−0.59,&amp;nbsp;∞)x=0,&amp;nbsp;f&amp;nbsp;″(0)&amp;gt;0If&amp;nbsp;f&amp;nbsp;″(x)&amp;gt;0⇒f(x)&amp;nbsp;is convexIf&amp;nbsp;f&amp;nbsp;″(x)&amp;lt;0⇒f(x)&amp;nbsp;is concaveconvex interval:&amp;nbsp;(−∞,&amp;nbsp;−3.41)∪(−0.59,&amp;nbsp;∞)concave interval:&amp;nbsp;(−3.41,&amp;nbsp;−0.59)

How do you determine whether the function

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