Let f and g be two differentiable functins such that:
`f (x)=g '(1) sin x+ (g'' (2) -1) x`
`g (x) = x^(2) -f'((pi)/(2)) x+ f'(-(pi)/(2))`
The number of solution (s) of the equation `f (x) = g (x)` is/are :

Options

A. 1
B. 2
C. 3
D. infinite

Answer

To solve the problem, we need to find the number of solutions to the equation \( f(x) = g(x) \) given the functions \( f(x) \) and \( g(x) \).

### Step-by-step Solution:

1. **Identify the Functions**:
We have:
\[ f(x) = g'(1) \sin x + (g''(2) - 1)x \]
\[ g(x) = x^2 - f'(\frac{\pi}{2}) x + f'(-\frac{\pi}{2}) \]

2. **Calculate \( g'(x) \)**:
To find \( g'(x) \), we differentiate \( g(x) \):
\[ g'(x) = \frac{d}{dx}(x^2 - f'(\frac{\pi}{2}) x + f'(-\frac{\pi}{2}) ) \]
\[ g'(x) = 2x - f'(\frac{\pi}{2}) \]

3. **Evaluate \( g'(1) \)**:
Substitute \( x = 1 \) into \( g'(x) \):
\[ g'(1) = 2(1) - f'(\frac{\pi}{2}) = 2 - f'(\frac{\pi}{2}) \]

4. **Calculate \( g''(x) \)**:
Differentiate \( g'(x) \) to find \( g''(x) \):
\[ g''(x) = \frac{d}{dx}(2x - f'(\frac{\pi}{2})) = 2 \]
Therefore,
\[ g''(2) = 2 \]

5. **Substitute \( g'(1) \) and \( g''(2) \) into \( f(x) \)**:
Now substitute back into \( f(x) \):
\[ f(x) = (2 - f'(\frac{\pi}{2})) \sin x + (2 - 1)x = (2 - f'(\frac{\pi}{2})) \sin x + x \]

6. **Calculate \( f'(\frac{\pi}{2}) \)**:
Differentiate \( f(x) \):
\[ f'(x) = (2 - f'(\frac{\pi}{2})) \cos x + 1 \]
Evaluate at \( x = \frac{\pi}{2} \):
\[ f'(\frac{\pi}{2}) = (2 - f'(\frac{\pi}{2})) \cos(\frac{\pi}{2}) + 1 = 0 + 1 = 1 \]

7. **Substitute \( f'(\frac{\pi}{2}) \) into \( g(x) \)**:
Now substitute \( f'(\frac{\pi}{2}) = 1 \) back into \( g(x) \):
\[ g(x) = x^2 - 1 \cdot x + 1 = x^2 - x + 1 \]

8. **Set up the equation \( f(x) = g(x) \)**:
Now we set \( f(x) = g(x) \):
\[ (2 - 1) \sin x + x = x^2 - x + 1 \]
Simplifying gives:
\[ \sin x + x = x^2 - x + 1 \]
Rearranging:
\[ \sin x = x^2 - 2x + 1 \]
Which simplifies to:
\[ \sin x = (x - 1)^2 \]

9. **Analyze the equation \( \sin x = (x - 1)^2 \)**:
The left side, \( \sin x \), oscillates between -1 and 1, while the right side, \( (x - 1)^2 \), is a parabola opening upwards with its vertex at \( (1, 0) \).

10. **Determine the number of intersections**:
- For \( x < 1 \), \( (x - 1)^2 \) is increasing from 0 to 1.
- For \( x = 1 \), both sides equal 0.
- For \( x > 1 \), \( (x - 1)^2 \) increases and will intersect \( \sin x \) again before \( x = 2 \).

By sketching or analyzing the graphs, we can see that there are exactly **two points of intersection**.

### Conclusion:
The number of solutions to the equation \( f(x) = g(x) \) is **2**.