What are the points of inflection of #f(x)=x^{2}e^{11 -x} #?

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Answer 1 · 2017-12-14T18:43:41

#f(x)=x^2e^(11−x)#

#f'(x)=2xe^(11−x)+x^2e^(11−x)*-1#

#f'(x)=e^(11−x)[2x-x^2]#

#f''(x)=-e^(11−x)(2x-x^2)+e^(11−x)(2-2x)#

#f''(x)=e^(11−x)[-2x+x^2+2-2x]#

#f''(x)=e^(11−x)[x^2-4x+2]#

#e^(11−x)>0quad# That means we don't care about that. BUT:

#x^2-4x+2=0#

#x_(1,2)=(-b+-sqrt(b^2-4ac))/(2a)#

#x_(1,2)=(4+-sqrt(4^2-4*1*2))/(2*1)#

#x_(1,2)=(4+-sqrt(16-8))/(2)=(2*2+-sqrt(2^2*2))/(2)#

#x_(1,2)=(cancel2*2+-cancel2sqrt(2))/(cancel2)=2+-sqrt(2)#

#x_1=2-sqrt(2)#

#x_2=2+sqrt(2)#