Minimum value of `4x^2-4x|sin theta|-cos^2 theta` is equal

Options

A. (a) `-2`
B. (b) `-1`
C. (c) `-(1)/(2)`
D. (d) `0`

Answer

To find the minimum value of the expression \(4x^2 - 4x|\sin \theta| - \cos^2 \theta\), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ E = 4x^2 - 4x|\sin \theta| - \cos^2 \theta \]

### Step 2: Identify it as a quadratic in \(x\)
This expression is a quadratic function in terms of \(x\). We can write it in standard form:
\[ E = 4(x^2 - x|\sin \theta|) - \cos^2 \theta \]

### Step 3: Complete the square
To find the minimum value of the quadratic \(4(x^2 - x|\sin \theta|)\), we can complete the square:
\[ x^2 - x|\sin \theta| = \left(x - \frac{|\sin \theta|}{2}\right)^2 - \frac{|\sin \theta|^2}{4} \]
Thus, we can rewrite \(E\):
\[ E = 4\left(\left(x - \frac{|\sin \theta|}{2}\right)^2 - \frac{|\sin \theta|^2}{4}\right) - \cos^2 \theta \]
\[ E = 4\left(x - \frac{|\sin \theta|}{2}\right)^2 - |\sin \theta|^2 - \cos^2 \theta \]

### Step 4: Simplify the expression
Notice that \( |\sin \theta|^2 + \cos^2 \theta = 1 \):
\[ E = 4\left(x - \frac{|\sin \theta|}{2}\right)^2 - 1 \]

### Step 5: Determine the minimum value
The term \(4\left(x - \frac{|\sin \theta|}{2}\right)^2\) is always non-negative and achieves its minimum value of \(0\) when \(x = \frac{|\sin \theta|}{2}\). Therefore, the minimum value of \(E\) occurs when:
\[ E_{\text{min}} = 0 - 1 = -1 \]

### Conclusion
Thus, the minimum value of the expression \(4x^2 - 4x|\sin \theta| - \cos^2 \theta\) is:
\[ \boxed{-1} \]