Answer:
The equation of the tangent line for the function
at
is ![y=\frac{4}{e^{2}}](https://tex.z-dn.net/?f=y%3D%5Cfrac%7B4%7D%7Be%5E%7B2%7D%7D)
Step-by-step explanation:
To find the tangent line to
at ![x_0=2](https://tex.z-dn.net/?f=x_0%3D2)
Firstly, find the slope of the tangent line, which is the derivative of the function, evaluated at the point: ![m=f^{\prime}\left(2\right)](https://tex.z-dn.net/?f=m%3Df%5E%7B%5Cprime%7D%5Cleft%282%5Cright%29)
![\mathrm{Apply\:the\:Product\:Rule}:\quad \left(f\cdot g\right)'=f\:'\cdot g+f\cdot g'\\\\f=x^2,\:g=e^{-x}](https://tex.z-dn.net/?f=%5Cmathrm%7BApply%5C%3Athe%5C%3AProduct%5C%3ARule%7D%3A%5Cquad%20%5Cleft%28f%5Ccdot%20g%5Cright%29%27%3Df%5C%3A%27%5Ccdot%20g%2Bf%5Ccdot%20g%27%5C%5C%5C%5Cf%3Dx%5E2%2C%5C%3Ag%3De%5E%7B-x%7D)
![\frac{d}{dx}\left(x^2e^{-x}\right)=\frac{d}{dx}\left(x^2\right)e^{-x}+\frac{d}{dx}\left(e^{-x}\right)x^2\\\\=2e^{-x}x-e^{-x}x^2](https://tex.z-dn.net/?f=%5Cfrac%7Bd%7D%7Bdx%7D%5Cleft%28x%5E2e%5E%7B-x%7D%5Cright%29%3D%5Cfrac%7Bd%7D%7Bdx%7D%5Cleft%28x%5E2%5Cright%29e%5E%7B-x%7D%2B%5Cfrac%7Bd%7D%7Bdx%7D%5Cleft%28e%5E%7B-x%7D%5Cright%29x%5E2%5C%5C%5C%5C%3D2e%5E%7B-x%7Dx-e%5E%7B-x%7Dx%5E2)
Next, evaluate the derivative at the given point to find the slope.
![\mathrm{Plug\:}x=2\mathrm{\:into\:the\:equation\:}2e^{-x}x-e^{-x}x^2\\\\2e^{-2}\cdot \:2-e^{-2}\cdot \:2^2=0](https://tex.z-dn.net/?f=%5Cmathrm%7BPlug%5C%3A%7Dx%3D2%5Cmathrm%7B%5C%3Ainto%5C%3Athe%5C%3Aequation%5C%3A%7D2e%5E%7B-x%7Dx-e%5E%7B-x%7Dx%5E2%5C%5C%5C%5C2e%5E%7B-2%7D%5Ccdot%20%5C%3A2-e%5E%7B-2%7D%5Ccdot%20%5C%3A2%5E2%3D0)
![m=f^{\prime}\left(2\right)=0](https://tex.z-dn.net/?f=m%3Df%5E%7B%5Cprime%7D%5Cleft%282%5Cright%29%3D0)
Finally, the equation of the tangent line is ![y-y_0=m(x-x_0)](https://tex.z-dn.net/?f=y-y_0%3Dm%28x-x_0%29)
Plugging the found values, we get that
![y-\left(\frac{4}{e^{2}}\right)=0\left(x-\left(2\right)\right)\\\\y=\frac{4}{e^{2}}](https://tex.z-dn.net/?f=y-%5Cleft%28%5Cfrac%7B4%7D%7Be%5E%7B2%7D%7D%5Cright%29%3D0%5Cleft%28x-%5Cleft%282%5Cright%29%5Cright%29%5C%5C%5C%5Cy%3D%5Cfrac%7B4%7D%7Be%5E%7B2%7D%7D)
The equation of the tangent line for the function
at
is ![y=\frac{4}{e^{2}}](https://tex.z-dn.net/?f=y%3D%5Cfrac%7B4%7D%7Be%5E%7B2%7D%7D)