Applying derivatives, it is found that the correct options are:
A. [tex]y^{\prime\prime}(x) = -\sin{x} + e^{-x}[/tex]
D. [tex]y - y^{\prime\prime\prime\prime}(x) = 3x[/tex]
The function given is:
[tex]f(x) = \sin{x} + e^{-x} + 3x[/tex]
It's first four derivatives are:
[tex]y^{\prime}(x) = \cos{x} - e^{-x} + 3[/tex]
[tex]y^{\prime\prime}(x) = -\sin{x} + e^{-x}[/tex]
Hence, option A is correct.
[tex]y^{\prime\prime\prime}(x) = -cos{x} - e^{-x}[/tex]
[tex]y^{\prime\prime\prime}(x) \neq y^{\prime}(x)[/tex], hence, option B is incorrect.
[tex]y^{\prime\prime\prime\prime}(x) = sin{x} + e^{-x}[/tex]
It is not the multiplication of the first and of the third derivative, hence option C is incorrect.
[tex]y - y^{\prime\prime\prime\prime}(x) = 3x[/tex]
[tex]\sin{x} + e^{-x} + 3x - \sin{x} - e^{-x} = 3x[/tex]
[tex]3x = 3x[/tex]
Hence, option D is correct.
A similar problem is given at https://brainly.com/question/17042788
