dont understand this question (1 Viewer)

[iloveschool1996]

would be appreciated if someone helped me out here
http://i.imgur.com/pMbVrVX.jpg
the last question
cheers

 

[rumbleroar]

integrating f"(x) gives us f'(x) + c

f'(x) = -2/x + e^(2x) + c
f'(1) = e^2

therefore, f'(1) = -2/1 + e^2 + c = e^2
thus, c = 2 and f'(x) = -2/x + e^(2x) + 2

for f(x), integrating f'(x) gives us f(x) + c

f(x) = -2ln|x| + e^(2x)/2 + 2x + c
f(1) = e^2/2

i.e. f(1) = -2ln(1) + e^2/2 + 2 + c = e^2/2
therefore, c = -2

therefore, f(x) = -2ln|x| + e^(2x)/2 + 2x - 2