F(x)=x 2 +xg'(1)+g''(2) And G(x)=f(1)x 2 +xf'(x)+f'(x) Then The Value Of F(3)

To find the value of f(3) given the functions f(x) and g(x), we first need to analyze the expressions provided. Let's break down the functions step by step to derive the necessary values.

Understanding the Functions

We have two functions defined as follows:

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

To find f(3), we need to evaluate f(x) at x = 3. However, we first need to determine the values of g'(1), g''(2), and f(1) to fully understand f(x).

Finding f(1)

Let's start by calculating f(1):

• f(1) = 1² + 1g'(1) + g''(2) = 1 + g'(1) + g''(2)

At this point, we need the values of g'(1) and g''(2) to proceed. However, these derivatives depend on the function g(x), which is defined in terms of f(x).

Analyzing g(x)

Now, let's look at g(x):

• g(x) = f(1)x² + xf'(x) + f'(x)

To evaluate g'(1) and g''(2), we need to differentiate g(x). Let's find g'(x):

• g'(x) = d/dx [f(1)x²] + d/dx [xf'(x)] + d/dx [f'(x)]
• g'(x) = 2f(1)x + (f'(x) + xf''(x)) + f''(x)

Now, substituting x = 1 into g'(x):

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

Finding g''(x)

Next, we differentiate g'(x) to find g''(x):

• g''(x) = d/dx [2f(1)x + f'(x) + xf''(x) + f''(x)]
• g''(x) = 2f(1) + f''(x) + f''(x) + xf'''(x)

Now, substituting x = 2 into g''(x):

• g''(2) = 2f(1) + f''(2) + 2f'''(2)

Substituting Back into f(x)

Now that we have expressions for g'(1) and g''(2), we can substitute them back into f(1):

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

This equation is complex and requires specific values for f'(1), f''(1), f''(2), and f'''(2) to solve. However, we can simplify our approach by assuming certain values or relationships between these derivatives.

Finding f(3)

Once we have f(1), we can substitute it back into the original equation for f(x) to find f(3):

• f(3) = 3² + 3g'(1) + g''(2)
• f(3) = 9 + 3(2f(1) + f'(1) + 2f''(1)) + (2f(1) + f''(2) + 2f'''(2))

In conclusion, without specific values for the derivatives of f, we cannot compute an exact numerical value for f(3). However, the process outlined provides a structured approach to evaluate f(3) once those values are known. If you have any specific values for f'(1), f''(1), f''(2), or f'''(2), we can plug them in to find the exact value of f(3).