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

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maggionmoo · Accepted Answer

Step 1f(x)=xe−xf′(x)=(1)e−x+x[e−x(−1)]=e−x−xe−x=−e−x(x−1)So, f″(x)=[−e−x(−1)](x−1)+(−e−x)(1)=e−x(x−1)−e−x=e−x(x−2)Now, fx)=e−x(x−2) is continuous on its domain, (−∞,∞), so the only way it can change sign is by passing through zero. (The only partition numbers are the zeros of f&quot;(x))Step 2fx)=0 if and only if either e−x=0 or x−2=0e to any (real) power is positive, so the only way for f&quot; to be 0 is for x to be 2.We partition the number line:(−∞,2) and (2,∞)On the interval (−∞,2), we have fx)&amp;lt;0 so f is concave down.On (2,∞), we have fx)&amp;gt;0, so f is concave up.Inflection pointThe point (2,f(2))=(2,2e2) is the only inflection point for the graph of this function.

How do you determine whether the function

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