Find `max^(m)&min^(m)` value of `32sin^(2)alpha+43cos^(2)alpha`.

Answer

To find the maximum and minimum values of the expression \( 32 \sin^2 \alpha + 43 \cos^2 \alpha \), we can follow these steps:

### Step 1: Rewrite the Expression
We start with the expression:
\[ 32 \sin^2 \alpha + 43 \cos^2 \alpha \]
We can rewrite \( \cos^2 \alpha \) in terms of \( \sin^2 \alpha \):
\[ \cos^2 \alpha = 1 - \sin^2 \alpha \]
Substituting this into the expression gives:
\[ 32 \sin^2 \alpha + 43 (1 - \sin^2 \alpha) = 32 \sin^2 \alpha + 43 - 43 \sin^2 \alpha \]
This simplifies to:
\[ (32 - 43) \sin^2 \alpha + 43 = -11 \sin^2 \alpha + 43 \]

### Step 2: Identify the Range of \( \sin^2 \alpha \)
The term \( \sin^2 \alpha \) ranges from 0 to 1, since \( \sin^2 \alpha \) can take any value between 0 and 1 inclusive.

### Step 3: Find Maximum and Minimum Values
Now we can evaluate the expression \( -11 \sin^2 \alpha + 43 \) at the endpoints of the range of \( \sin^2 \alpha \).

1. **When \( \sin^2 \alpha = 0 \)**:
\[ -11(0) + 43 = 43 \]

2. **When \( \sin^2 \alpha = 1 \)**:
\[ -11(1) + 43 = -11 + 43 = 32 \]

### Step 4: Conclusion
From the evaluations above, we find:
- The **maximum value** of the expression is \( 43 \).
- The **minimum value** of the expression is \( 32 \).

Thus, the maximum and minimum values are:
\[ \text{max} = 43, \quad \text{min} = 32 \]