Answer to Question #155252 in Calculus for Tom Garland

Question #155252

Find all intervals on which the function f(x)=(-x+2)e^-x is increasing. If there is no such interval, submit an empty answer.


1
Expert's answer
2021-01-13T19:16:30-0500

We have function "f(x) = (-x+2)*e^{-x}" and we must find interval, where it is increasing. Firstly, let's find the derivative of function:

"f'(x) = -e^{-x} + (-x)(-1)e^{-x} - 2e^{-x}=-3e^{-x} + xe^{-x} ="

"= e^{-x} (x-3)"

We know that function "e^{-x}" is always positive function and descending function, so "e^{-x} >0" for "\\forall x\\in R"

For increase function f(x) we must find inteval, where "f'(x) \\ge0"

"e^{-x} (x-3) \\ge 0 \\implies x-3 \\ge 0 \\implies x \\ge 3"

So we have answer "x\\in[3; +\\infty)", where function is increasing


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