To determine

To investigate: the family of curves f(x)=xne−x

Answer

The family of curves f(x)=xne−x have no inflection points for any values of n.

Explanation

Members of the family of curves f(x)=xne−x is depicted below, we can observe that

f(x) Increases for even values and decreases for odd values

For n=0 Forn=−2

For n=1 Forn=2

limx→∞f(x)=limx→∞xne−x=limx→∞xnex (∞∞Form)Applying l'Hospital's rule, we get=limx→∞nxn−1ex (∞∞Form)we observe that it is in indeterminate form, So we apply l'Hospital's Rule repeatedly=limx→∞n!x0ex=limx→∞n!ex=n!e∞=0

For maxima/minima f′(x)=0

f(x)=xne−x

Differentiating with respect to x

f′(x)=xn(−e−x)+nxn−1(e−x)=e−xxn−1(−x+n)f′(x)=0⇒e−xxn−1(−x+n)=0⇒xn−1(−x+n)=0 (e−x≠0)⇒x=0  or x=n

The maxima of the function f(x)=xne−x occurs at x=n( n is odd) and minima at the point x=0(n is even)

The family of curves f(x)=xne−x have no inflection points for any values of n.

Conclusion: The local maxima of the function f(x)=xne−x occurs at x=n( n is odd) and local minima at the point x=0(n is even)

The family of curves f(x)=xne−x have no inflection points for any values of n.