The greatest value of 5sin2x+7cos2x-4sinxcosx will be:

Question

The greatest value of $5 \sin^{2} x + 7 \cos^{2} x - 4 \sin x \cos x$ will be:

Smart Achievers · Accepted Answer

(2)

Ans. $6 + \sqrt{5}$

Explanation:

$y = 5 \sin^{2} x + 7 \cos^{2} x - 4 \sin x \cos x$

$= {(2 \sin x - \cos x)}^{2} + \sin^{2} x + 6 \cos^{2} x$

$= {(2 \sin x - \cos x)}^{2} + (\sin^{2} x + \cos^{2} x) + 5 \cos^{2} x$

$= {(2 \sin x - \cos x)}^{2} + 1 + 5 \cos^{2} x$

$y_{\max} = \sqrt{2^{2} + 1^{2}} + 1 + 5 = 6 + \sqrt{5}$

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