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

1 Answer
Dec 14, 2017

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)