To find the derivatives we have to use the following property:
[tex]\frac{d(uv)}{dx}=u^{\prime}v+uv^{\prime}[/tex]
Then the first derivative will be:
[tex]y\text{ = 2xe}^{-x}[/tex]
[tex]y^{\prime}\text{ = }2\mleft(e^{-x}-e^{-x}x\mright)[/tex]
Now to find the second derivative we have to use the same derivative property with the part "-xe^-x. Thus:
[tex]y^{\doubleprime}=2(-2e^{-x}+xe^{-x})[/tex]
Now we have to equal the equation to zero:
[tex]2(-2e^{-x}+xe^{-x})=0[/tex]
[tex]-2e^{-x}+xe^{-x}=0[/tex]
[tex]e^{-x}(x-2)=0_{}[/tex]
The first solution will be:
[tex]e^{-x}=0[/tex]
[tex]-x\text{ = ln 0}[/tex]
ln 0 is undefined, so this answer is impossible.
The second solution will be:
[tex]x-2\text{ = 0}[/tex]
[tex]x\text{ = 2}[/tex]
Answer: x = 2.
