Find `max^(m)&min^(m)` value of `10sin^(2)alpha+23cos^(2)alpha`.

Answer

To find the maximum and minimum values of the expression \(10\sin^2\alpha + 23\cos^2\alpha\), we can follow these steps:

### Step 1: Rewrite the Expression
We start with the expression:
\[ y = 10\sin^2\alpha + 23\cos^2\alpha \]
Using the identity \(\sin^2\alpha + \cos^2\alpha = 1\), we can express \(\sin^2\alpha\) in terms of \(\cos^2\alpha\):
\[ \sin^2\alpha = 1 - \cos^2\alpha \]
Substituting this into the expression gives:
\[ y = 10(1 - \cos^2\alpha) + 23\cos^2\alpha \]
This simplifies to:
\[ y = 10 - 10\cos^2\alpha + 23\cos^2\alpha \]
Combining like terms, we have:
\[ y = 10 + 13\cos^2\alpha \]

### Step 2: Determine the Range of \(\cos^2\alpha\)
The value of \(\cos^2\alpha\) ranges from 0 to 1, since \(\cos^2\alpha\) cannot be negative and cannot exceed 1.

### Step 3: Find Minimum Value
To find the minimum value of \(y\), we substitute the minimum value of \(\cos^2\alpha\):
\[ \text{Minimum value of } y = 10 + 13(0) = 10 \]

### Step 4: Find Maximum Value
To find the maximum value of \(y\), we substitute the maximum value of \(\cos^2\alpha\):
\[ \text{Maximum value of } y = 10 + 13(1) = 10 + 13 = 23 \]

### Conclusion
Thus, the minimum value of \(10\sin^2\alpha + 23\cos^2\alpha\) is 10, and the maximum value is 23.

### Final Answer
\[ \text{max}(y) = 23, \quad \text{min}(y) = 10 \]