STATEMENT - 1 : Let f be a twice differentiable function such that `f'(x) = g(x)` and `f''(x) = - f (x)` . If `h'(x) = [f(x)]^(2) + [g (x)]^(2) , h(1) = 8 and h (0) =2 Rightarrow h(2) =14` and STATEMENT - 2 : `h''(x)=0 `

Answer

To solve the problem, we will analyze the given statements step by step.

### Step 1: Understanding the given functions
We have:
- \( f'(x) = g(x) \)
- \( f''(x) = -f(x) \)

This implies that \( g'(x) = f''(x) = -f(x) \).

### Step 2: Finding \( h'(x) \)
The function \( h(x) \) is defined as:
\[ h'(x) = f(x)^2 + g(x)^2 \]

### Step 3: Finding \( h''(x) \)
To find \( h''(x) \), we differentiate \( h'(x) \):
\[ h''(x) = \frac{d}{dx}(f(x)^2 + g(x)^2) \]
Using the chain rule:
\[ h''(x) = 2f(x)f'(x) + 2g(x)g'(x) \]
Substituting \( g(x) = f'(x) \) and \( g'(x) = -f(x) \):
\[ h''(x) = 2f(x)g(x) + 2g(x)(-f(x)) \]
This simplifies to:
\[ h''(x) = 2f(x)g(x) - 2f(x)g(x) = 0 \]
Thus, we conclude that:
\[ h''(x) = 0 \]

### Step 4: Analyzing Statement 2
Since we have found that \( h''(x) = 0 \), Statement 2 is **true**.

### Step 5: Finding the form of \( h(x) \)
Since \( h''(x) = 0 \), \( h(x) \) must be a linear function:
\[ h(x) = kx + c \]

### Step 6: Using the given values to find \( k \) and \( c \)
We know:
- \( h(1) = 8 \)
- \( h(0) = 2 \)

From \( h(0) = 2 \):
\[ c = 2 \]

From \( h(1) = 8 \):
\[ k(1) + c = 8 \implies k + 2 = 8 \implies k = 6 \]

Thus, we have:
\[ h(x) = 6x + 2 \]

### Step 7: Finding \( h(2) \)
Now, we can find \( h(2) \):
\[ h(2) = 6(2) + 2 = 12 + 2 = 14 \]

### Conclusion
Both statements can be evaluated:
- Statement 1: \( h(2) = 14 \) is **true**.
- Statement 2: \( h''(x) = 0 \) is also **true**.

### Final Answer
- Statement 1 is true.
- Statement 2 is true.