First slide

Maxima and minima

Question

$The maximum value of f (x) = |\begin{matrix} \sin^{2} x & 1 + \cos^{2} x & \cos 2 x \\ 1 + \sin^{2} x & \cos^{2} x & \cos 2 x \\ \sin^{2} x & \cos^{2} x & \sin 2 x \end{matrix}|, x \in R is:$

Moderate

A

5
B

$\sqrt{7}$
C

$\frac{3}{4}$
D

$\sqrt{5}$

Solution

$\begin{array}{l} T h e g i v e n \det e r \min e n t f (x) = |\begin{matrix} \sin^{2} x & 1 + \cos^{2} x & c o x 2 x \\ 1 + \sin^{2} x & \cos^{2} x & c o x 2 x \\ \sin^{2} x & \cos^{2} x & \sin 2 x \end{matrix}| \\ = |\begin{matrix} 2 & 1 + \cos^{2} x & c o x 2 x \\ 2 & \cos^{2} x & c o x 2 x \\ 1 & \cos^{2} x & \sin 2 x \end{matrix}| C_{1} \to C_{1} + C_{2} \\ = |\begin{matrix} 0 & 1 & 0 \\ 2 & \cos^{2} x & c o x 2 x \\ 1 & \cos^{2} x & \sin 2 x \end{matrix}| R_{1} \to R_{1} - R_{2} \\ E x p a n d t h e \det e r \min a n t \\ f (x) = 2 \sin 2 x - \cos 2 x \\ T h e \max i m u m v a l u e o f a \cos x + b \sin x i s \sqrt{a^{2} + b^{2}} \\ H e n c e \max i m u m v a l u e o f f (x) = \sqrt{2^{2} + 1} = \sqrt{5} \end{array}$

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