K(x) = ((3 cos x)/(5+4sin x)) {: ((max K(x))),((min K(x))) } =?


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Question Number 163700 by blackmamba last updated on 09/Jan/22

$K (x) = \frac{3 \cos x}{5 + 4 \sin x}$ $\begin{matrix} m a x K (x) \\ m i n K (x) \end{matrix}} = ?$
	Answered by mr W last updated on 10/Jan/22

	$\frac{3 \cos x}{5 + 4 \sin x} = k$ $3 \cos x - 4 k \sin x = 5 k$ $\frac{3}{\sqrt{3^{2} + {(4 k)}^{2}}} \cos x - \frac{4 k}{\sqrt{3^{2} + {(4 k)}^{2}}} \sin x = \frac{5 k}{\sqrt{3^{2} + {(4 k)}^{2}}}$ $\cos α \cos x - \sin α \sin x = \frac{5 k}{\sqrt{3^{2} + {(4 k)}^{2}}}$ $\cos (x + α) = \frac{5 k}{\sqrt{3^{2} + {(4 k)}^{2}}}$ $\frac{5 k}{\sqrt{3^{2} + {(4 k)}^{2}}} ⩽ 1$ $25 k^{2} ⩽ 9 + 16 k^{2}$ $k^{2} ⩽ 1$ $- 1 ⩽ k ⩽ 1$ $k {(x)}_{m i n} = - 1$ $k {(x)}_{m a x} = 1$
	Answered by MJS_new last updated on 09/Jan/22

	$\frac{3 c}{5 + 4 s} = \pm \frac{3 \sqrt{1 - s^{2}}}{5 + 4 s}$ $\pm \frac{d}{d s} [\frac{3 \sqrt{1 - s^{2}}}{5 + 4 s}] = 0$ $\mp \frac{3 (5 s + 4)}{{(4 s + 5)}^{2} \sqrt{1 - s^{2}}} = 0 \Rightarrow s = - \frac{4}{5} \Rightarrow - 1 ⩽ K ⩽ 1$
	Commented by blackmamba last updated on 09/Jan/22

	$y e s$
	Answered by cortano1 last updated on 09/Jan/22
	$(Q) Given K(x) = ((3cos x)/(5+4sin x)) Find max & min of K(x) Let y=K(x)=((3cos x)/(5+4sin x)) 5y+4y sin x=3cos x 5y = 3cos x−4y sin x ...(i) By Cauchy − Schwarz ∣ 3cos x−4y sin x ∣ ≤ (√((3)^2 +(−4y)^2 )) ∣5y∣ ≤ (√(9+16y^2 )) ⇔ 25y^2 ≤ 9+16y^2 ⇔ 9(y^2 −1)≤ 0 ⇔ −1≤y≤1 ∴ −1≤K(x)≤1 ⇒ {: ((max K(x)=1)),((min K(x)=−1)) }$
	$(Q) G i v e n K (x) = \frac{3 \cos x}{5 + 4 \sin x}$ $F i n d m a x & m i n o f K (x)$ $L e t y = K (x) = \frac{3 \cos x}{5 + 4 \sin x}$ $5 y + 4 y \sin x = 3 \cos x$ $5 y = 3 \cos x - 4 y \sin x . . . (i)$ $B y C a u c h y - S c h w a r z$ $∣ 3 \cos x - 4 y \sin x ∣ ⩽ \sqrt{{(3)}^{2} + {(- 4 y)}^{2}}$ $∣ 5 y ∣ ⩽ \sqrt{9 + 16 y^{2}}$ $\Leftrightarrow 25 y^{2} ⩽ 9 + 16 y^{2}$ $\Leftrightarrow 9 (y^{2} - 1) ⩽ 0$ $\Leftrightarrow - 1 ⩽ y ⩽ 1$ $∴ - 1 ⩽ K (x) ⩽ 1 \Rightarrow \begin{matrix} m a x K (x) = 1 \\ m i n K (x) = - 1 \end{matrix}}$
	Commented by blackmamba last updated on 09/Jan/22

	$y e s$
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