f(x)=x2+xg′(1)+g′′(2) and g(x)=f(1)x2+xf′(x)+f′′(x)

Solution:

Given:

f(x) = x^2 + xg'(1) + g''(2)

g(x) = f(1)x^2 + xf'(x) + f''(x)

First, calculate f(1):

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

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

Next, find the derivatives of g(x):

First derivative g'(x):

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

Second derivative g''(x):

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

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

Evaluate the derivatives at specific points:

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

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

Substitute these back into the original equation for f(x):

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

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

Combine like terms:

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

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

To solve this system, additional information about the specific forms or values of the derivatives f'(1), f''(1), and f''(2) is required.