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Question

Let f(x)=x2+xg'(1)+g''(2) and g(x)=x2+xf'(2)+f''(3), then:


A

f'(1)=4+f'(2)

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B

g'(2)=g'(1)+4

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C

g''(2)+f'(3)=8

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D

None of these

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Solution

The correct option is D

None of these


Explanation for the correct answer

Given that: f(x)=x2+xg'(1)+g''(2) and g(x)=x2+xf'(2)+f''(3)

f(x)=x2+xg'(1)+g''(2)f'(x)=2x+g'(1)....(1)

g(x)=x2+xf'(2)+f''(3)g'(x)=2x+f'(2).....(2)

Putting x=1 in equations (1) and (2):

f'(1)=2+g'(1)g'(1)=2+f'(2)

Putting x=2 in equation (1);

f'(2)=4+g'(1)g'(2)=4+f'(2)

Then,

g'(2)=4+4+g'(1)g'(2)=8+g'(1)

Also,

f''(x)=2,g''(x)=2f''(3)=2,g''(3)=2g''(x)+f''(x)=2+2=4

Hence, the correct answer is Option (D).


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