Let`f(x)=x^(2)+xg'(1)+g''(2) and g(x)=x^(2)+xf'(2)+f''(3).` Then

Options

A. `f'(1)=4+f'(2)`
B. `g'(2)=8+g0(1)`
C. `g''(2)+f''(3)=4`
D. all the above

Answer

D

Explanation

We have,
`f(x)=x^(2)+xg'(1)+g''(2)" and "g(x)=x^(2)+xf'(2)+f''(3)`
`implies" "f'(x)=2x+g'(1)" and "g'(x)=2x+f'(2)" "...(i)`
Putting x=1 in (i), we get
`f'(1)=2+g'(1)" and "g'(1)=2+f'(2)`
`implies" "f'(1)=4+f'(2)`
Putting x=2 in (i), we get
`f'(2)=4+g'(1)" and "g'(2)=4+f'(2)`
`implies" "g'(2)=4+4+g'(1)=8+g'(1)`
Ditterentiating (i) w.r.t. x, we get
`f''(x)=2" and "g''(x)=2" for all "x`
`implies" "f''(3)=2" and "g''(2)=2`
`implies" "g''(2)+f''(3)=2+2=4.`