Find the least value of `2 sin^(2) theta+ 3 cos^(2) theta`.

Answer

To find the least value of the expression \( y = 2 \sin^2 \theta + 3 \cos^2 \theta \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ y = 2 \sin^2 \theta + 3 \cos^2 \theta \]
Using the Pythagorean identity, we know that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
We can express \( \sin^2 \theta \) in terms of \( \cos^2 \theta \):
\[ \sin^2 \theta = 1 - \cos^2 \theta \]
Substituting this into the expression for \( y \):
\[ y = 2(1 - \cos^2 \theta) + 3 \cos^2 \theta \]

### Step 2: Simplify the expression
Now, we simplify the expression:
\[ y = 2 - 2 \cos^2 \theta + 3 \cos^2 \theta \]
Combining like terms gives:
\[ y = 2 + \cos^2 \theta \]

### Step 3: Determine the range of \( \cos^2 \theta \)
The value of \( \cos^2 \theta \) ranges from 0 to 1, since \( \cos^2 \theta \) is always non-negative and cannot exceed 1.

### Step 4: Find the minimum value of \( y \)
To find the least value of \( y \), we consider the minimum value of \( \cos^2 \theta \):
\[ \text{Minimum value of } \cos^2 \theta = 0 \]
Substituting this into the expression for \( y \):
\[ y_{\text{min}} = 2 + 0 = 2 \]

### Conclusion
Thus, the least value of \( 2 \sin^2 \theta + 3 \cos^2 \theta \) is:
\[ \boxed{2} \]