Let `f(x)=3x^(2)+4xg'(1)+g''(2)`
and, `g(x)=2x^(2)+3xf'(2)+f''(3)" for all "x in R.` Then,

Options

A. `f'(1)=22+12f'(2)`
B. `g,(2)=44+12g'(1)`
C. `f''(3)+g''(2)=10`
D. all the above

Answer

D

Explanation

We have,
`f(x)=3x^(2)+4xg'(1)+g''(2)" "...(i)`
`g(x)=2x^(2)+3xf'(2)+f''(3)" "...(ii)`
`:." "f'(x)=6x+4g'(1)" "...(iii)`
and, `g'(x)=4x+3f'(2)" "...(iv)`
`implies" "f''(x)=6" "...(v)`
and, `g''(x)=4" "...(vi)`
Putting x=1 in (iii) and (iv), we get
`f'(1)=6+4g'(1)" and "g'(1)=4+3f'(2)`
`implies" "f'(1)=6+4{4+3f'(2)}`
`implies" "f'(1)=22+12f'(2)`
Putting x=2 in (iv) , we get
`g'(2)=8+3f'(2)`
`implies" "g'(2)=8+3{12+4g'(1)}[{:("Putting x=2 in (iii) we get"),(f'(2)=12+4g'(1)):}]`
`implies" "g'(2)=44+12g'(1)`
Putting x=3 in (v) and x=2 in (vi), we get
`f''(3)=6" and "g''(2)=4`
`implies" "f''(3)+g''(2)=6+4=10.`