Statement 1: The range of `f(x)=sin^(2)x-sinx+1` is `[3/4,oo)`.
Statemen 2: The range of `f(x)=x^(2)-x+1` is `[3/4,oo)AAxepsilonR`.

Options

A. Statement 1 is True, Statement 2 is True, Statement -2 is a correct explanation for Statement -1
B. Statement -1 is True, Statement -2 is True, Statement -2 is NOT a correct explanation for Statemnt -1
C. Statement -1 is True, Statement -2 is False
D. Statement -1 is False, Statement -2 is True

Explanation

To solve the problem, we will analyze both statements regarding the functions given.

### Statement 1:
**Function:** \( f(x) = \sin^2 x - \sin x + 1 \)

**Step 1:** Rewrite the function.
\[ f(x) = \sin^2 x - \sin x + 1 \]

**Step 2:** Let \( y = \sin x \). Then, we can rewrite the function in terms of \( y \):
\[ f(y) = y^2 - y + 1 \]
where \( y \) ranges from \(-1\) to \(1\) since \( \sin x \) can take any value in this interval.

**Step 3:** Find the range of \( f(y) \).
To find the minimum value of \( f(y) \), we can complete the square:
\[ f(y) = (y - \frac{1}{2})^2 + \frac{3}{4} \]

**Step 4:** Determine the minimum value.
The minimum value occurs when \( (y - \frac{1}{2})^2 = 0 \), which gives:
\[ f(y)_{\text{min}} = \frac{3}{4} \]

**Step 5:** Determine the maximum value.
Evaluate \( f(y) \) at the endpoints of the interval:
- For \( y = -1 \):
\[ f(-1) = (-1)^2 - (-1) + 1 = 1 + 1 + 1 = 3 \]
- For \( y = 1 \):
\[ f(1) = (1)^2 - (1) + 1 = 1 - 1 + 1 = 1 \]

Thus, the maximum value of \( f(y) \) is \( 3 \).

**Step 6:** Conclusion for Statement 1.
The range of \( f(x) \) is:
\[ \left[\frac{3}{4}, 3\right] \]
This contradicts the statement that the range is \(\left[\frac{3}{4}, \infty\right)\). Therefore, **Statement 1 is false**.

### Statement 2:
**Function:** \( g(x) = x^2 - x + 1 \)

**Step 1:** Rewrite the function.
\[ g(x) = x^2 - x + 1 \]

**Step 2:** Complete the square.
\[ g(x) = (x - \frac{1}{2})^2 + \frac{3}{4} \]

**Step 3:** Determine the minimum value.
The minimum value occurs when \( (x - \frac{1}{2})^2 = 0 \):
\[ g(x)_{\text{min}} = \frac{3}{4} \]

**Step 4:** Determine the maximum value.
As \( x \) approaches \( \infty \) or \( -\infty \), \( g(x) \) approaches \( \infty \).

**Step 5:** Conclusion for Statement 2.
The range of \( g(x) \) is:
\[ \left[\frac{3}{4}, \infty\right) \]
This confirms that **Statement 2 is true**.

### Final Conclusion:
- Statement 1 is false.
- Statement 2 is true.