`f(x)` and `g(x)` are two differentiable functions in `[0,2]` such that `f"(x)=g"(x)=0, f'(1)=2, g'(1)=4, f(2)=3, g(2)=9` then `f(x)-g(x)` at `x=3/2` is

Options

A. 0
B. 2
C. 10
D. 5

Explanation

To solve the problem, we need to find the value of \( f(x) - g(x) \) at \( x = \frac{3}{2} \). Given the conditions in the problem, we can follow these steps:

### Step 1: Analyze the given information
We know that:
- \( f''(x) = g''(x) = 0 \) for all \( x \) in the interval \([0, 2]\).
- This implies that both \( f(x) \) and \( g(x) \) are linear functions.

### Step 2: Write the general form of \( f(x) \) and \( g(x) \)
Since \( f''(x) = 0 \) and \( g''(x) = 0 \), we can express \( f(x) \) and \( g(x) \) as:
\[ f(x) = ax + b \]
\[ g(x) = cx + d \]
where \( a, b, c, d \) are constants.

### Step 3: Use the derivative information
From the problem, we have:
- \( f'(1) = 2 \) implies \( a = 2 \).
- \( g'(1) = 4 \) implies \( c = 4 \).

Thus, we can rewrite the functions as:
\[ f(x) = 2x + b \]
\[ g(x) = 4x + d \]

### Step 4: Use the function values at \( x = 2 \)
We also know:
- \( f(2) = 3 \) gives us:
\[ 2(2) + b = 3 \implies 4 + b = 3 \implies b = -1 \]
So, \( f(x) = 2x - 1 \).

- \( g(2) = 9 \) gives us:
\[ 4(2) + d = 9 \implies 8 + d = 9 \implies d = 1 \]
So, \( g(x) = 4x + 1 \).

### Step 5: Find \( f(x) - g(x) \)
Now we can find \( f(x) - g(x) \):
\[ f(x) - g(x) = (2x - 1) - (4x + 1) = 2x - 1 - 4x - 1 = -2x - 2 \]

### Step 6: Evaluate at \( x = \frac{3}{2} \)
Now we substitute \( x = \frac{3}{2} \):
\[ f\left(\frac{3}{2}\right) - g\left(\frac{3}{2}\right) = -2\left(\frac{3}{2}\right) - 2 = -3 - 2 = -5 \]

Thus, the value of \( f\left(\frac{3}{2}\right) - g\left(\frac{3}{2}\right) \) is \(-5\).

### Final Answer
\[ f\left(\frac{3}{2}\right) - g\left(\frac{3}{2}\right) = -5 \]

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