Let
, $x \in [0, \pi]$. Then the maximum value of $f(x)$ is equal to _________.
Let
, $x \in [0, \pi]$. Then the maximum value of $f(x)$ is equal to _________.
Show Hint
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.
Correct Answer: 6
Solution and Explanation
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 \(x=\pi\). \[ \boxed{\text{Maximum value of } f(x) = 6} \]
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