Question

Let 

, $x \in [0, \pi]$. Then the maximum value of $f(x)$ is equal to _________. 
 

Hint:

In determinants involving trigonometric entries:
First apply row/column operations to create zeros.
Convert everything into \(\sin^2 x\), \(\cos^2 x\), or \(\cos 2x\).
Reduce the determinant to a simple trigonometric expression before finding extrema.

Answer 1

Correct Answer: 6

Given, 

Step 1: Simplify using row operations 
Apply the row operation \(R_2 \to R_2 - R_1\): 


Step 3: Use trigonometric identities 
Recall: \[ \cos 2x = \cos^2 x - \sin^2 x \] So, \[ f(x) = 4 + 4\cos 2x - 2\cos 2x \] \[ \boxed{f(x)=4+2\cos 2x} \] Step 4: Find the maximum value 
Since \(x\in[0,\pi]\), we have \(2x\in[0,2\pi]\). \[ \max(\cos 2x)=1 \] Hence, \[ f_{\max}=4+2(1)=\boxed{6} \] This occurs at \(x=0\) or \(x=\pi\). \[ \boxed{\text{Maximum value of } f(x) = 6} \]