Answer:
Equation of tangent will be [tex]y=\frac{x}{e}[/tex]
Step-by-step explanation:
We have given the function [tex]y=x^2e^{-x}[/tex]
We have to find the equation of tangent at the point [tex](1,\frac{1}{e})[/tex]
Equation of tangent is equal to [tex]\frac{dy}{dx}[/tex]
So [tex]\frac{dy}{dx}=x^2\frac{d}{dx}e^{-x}+e^{-x}\frac{d}{dx}2x=-x^2e^{-x}+2e^{-x}[/tex]
Now we have given point [tex](1,\frac{1}{e})[/tex]
So putting these points in the equation of tangent
[tex]\frac{dy}{dx}=-1^2e^{-1}+2e^{-1}[/tex]
[tex]\frac{dy}{dx}=\frac{1}{e}[/tex]
Now equation of tangent passing through [tex](1,\frac{1}{e})[/tex]
[tex]y-\frac{1}{e}=\frac{1}{e}(x-1)[/tex]
[tex]y=\frac{x}{e}-\frac{1}{e}+\frac{1}{e}=\frac{x}{e}[/tex]
