Let f(x) = <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|"><mtable><mtr><mtd><mn>1</mn><mo>+</mo><msup><mi>sin</mi><mn>2</mn></msup><mi>x</mi></mtd><mtd><msup><mi>cos</mi><mn>2</mn></msup><mi>x</mi></mtd><mtd><mn>4</mn><mi>sin</mi><mn>2</mn><mi>x</mi></mtd></mtr><mtr><mtd><msup><mi>sin</mi><mn>2</mn></msup><mi>x</mi></mtd><mtd><mn>1</mn><mo>+</mo><msup><mi>cos</mi><mn>2</mn></msup><mi>x</mi></mtd><mtd><mn>4</mn><mi>sin</mi><mn>2</mn><mi>x</mi></mtd></mtr><mtr><mtd><msup><mi>sin</mi><mn>2</mn></msup><mi>x</mi></mtd><mtd><msup><mi>cos</mi><mn>2</mn></msup><mi>x</mi></mtd><mtd><mn>1</mn><mo>+</mo><mn>4</mn><mi>sin</mi><mn>2</mn><mi>x</mi></mtd></mtr></mtable></mfenced></math>, then the maximum value of f(x), is-
Step-by-Step Solution
Key Concept: Apply row operations R1 -> R1 - R3 and R2 -> R2 - R3 to simplify the determinant.
Applying R1 -> R1 - R3 and R2 -> R2 - R3, we get f(x) = |(1, 0, -1), (0, 1, -1), (sin^2 x, cos^2 x, 1+4sin 2x)|. Expanding along the first row: 1(1+4sin 2x + cos^2 x) - 0 + (-1)(0 - sin^2 x) = 1 + 4sin 2x + cos^2 x + sin^2 x = 1 + 4sin 2x + 1 = 2 + 4sin 2x. The maximum value of sin 2x is 1, so the maximum value of f(x) is 2 + 4(1) = 6.
Correct Answer: C