Maximum and minimum value of `2sin^(2)theta-3sintheta+2` is-

Options

A. `1/4, -7/4`
B. `1/4, 21/4`
C. `21/4, -3/4`
D. `7, 7/8`

Answer

To find the maximum and minimum values of the expression \( y = 2\sin^2\theta - 3\sin\theta + 2 \), we can follow these steps:

### Step 1: Rewrite the expression
Let \( x = \sin\theta \). Then we can rewrite the expression as:
\[ y = 2x^2 - 3x + 2 \]

**Hint:** Substitute \( \sin\theta \) with a variable to simplify the expression.

### Step 2: Complete the square
To find the maximum and minimum values, we can complete the square for the quadratic expression:
\[ y = 2(x^2 - \frac{3}{2}x) + 2 \]
Now, we need to complete the square for \( x^2 - \frac{3}{2}x \):
\[ x^2 - \frac{3}{2}x = \left(x - \frac{3}{4}\right)^2 - \frac{9}{16} \]
Substituting this back into the equation gives:
\[ y = 2\left(\left(x - \frac{3}{4}\right)^2 - \frac{9}{16}\right) + 2 \]
\[ y = 2\left(x - \frac{3}{4}\right)^2 - \frac{9}{8} + 2 \]
\[ y = 2\left(x - \frac{3}{4}\right)^2 + \frac{7}{8} \]

**Hint:** Completing the square helps to express the quadratic in a form that makes it easier to identify maximum and minimum values.

### Step 3: Analyze the completed square
The term \( 2\left(x - \frac{3}{4}\right)^2 \) is always non-negative (i.e., \( \geq 0 \)). Therefore, the minimum value of \( y \) occurs when \( \left(x - \frac{3}{4}\right)^2 = 0 \):
\[ y_{\text{min}} = \frac{7}{8} \]

**Hint:** The minimum value occurs when the squared term is zero.

### Step 4: Determine the maximum value
The maximum value of \( y \) occurs when \( x = \sin\theta \) is at its maximum, which is 1:
\[ y = 2(1)^2 - 3(1) + 2 = 2 - 3 + 2 = 1 \]

**Hint:** Substitute the maximum value of \( \sin\theta \) to find the maximum value of \( y \).

### Step 5: Conclusion
Thus, the minimum value of \( y \) is \( \frac{7}{8} \) and the maximum value of \( y \) is \( 1 \).

**Final Answer:** The maximum value is \( 1 \) and the minimum value is \( \frac{7}{8} \).