what-is-maximum-value-of-y-3x-4-2-sin-2x-

Question Number 77574 by jagoll last updated on 08/Jan/20

$${what}\:{is}\:{maximum}\: \\ $$$${value}\:{of}\:{y}=\:\left(\mathrm{3}{x}+\mathrm{4}\right)^{\mathrm{2}} ×\mathrm{sin}\:\left(\mathrm{2}{x}\right) \\ $$

Commented by MJS last updated on 08/Jan/20

sin 2x =−1 ⇒ x=nπ−(π/4)=((4n−1)/4)π sin 2x =+1 ⇒ x=nπ+(π/4)=((4n+1)/4)π n∈Z f(x)=(3x+4)^2 sin 2x f(((4n−1)/4)π)=−(((12πn−3π+16)^2 )/(16)) n=±10^3 ⇒ f(x)≈−8.9×10^7 n=±10^6 ⇒ f(x)≈−8.9×10^(13) n=±10^9 ⇒ f(x)≈−8.9×10^(19) ...

$$\mathrm{sin}\:\mathrm{2}{x}\:=−\mathrm{1}\:\Rightarrow\:{x}={n}\pi−\frac{\pi}{\mathrm{4}}=\frac{\mathrm{4}{n}−\mathrm{1}}{\mathrm{4}}\pi \\ $$$$\mathrm{sin}\:\mathrm{2}{x}\:=+\mathrm{1}\:\Rightarrow\:{x}={n}\pi+\frac{\pi}{\mathrm{4}}=\frac{\mathrm{4}{n}+\mathrm{1}}{\mathrm{4}}\pi \\ $$$${n}\in\mathbb{Z} \\ $$$$ \\ $$$${f}\left({x}\right)=\left(\mathrm{3}{x}+\mathrm{4}\right)^{\mathrm{2}} \mathrm{sin}\:\mathrm{2}{x} \\ $$$${f}\left(\frac{\mathrm{4}{n}−\mathrm{1}}{\mathrm{4}}\pi\right)=−\frac{\left(\mathrm{12}\pi{n}−\mathrm{3}\pi+\mathrm{16}\right)^{\mathrm{2}} }{\mathrm{16}} \\ $$$${n}=\pm\mathrm{10}^{\mathrm{3}} \:\Rightarrow\:{f}\left({x}\right)\approx−\mathrm{8}.\mathrm{9}×\mathrm{10}^{\mathrm{7}} \\ $$$${n}=\pm\mathrm{10}^{\mathrm{6}} \:\Rightarrow\:{f}\left({x}\right)\approx−\mathrm{8}.\mathrm{9}×\mathrm{10}^{\mathrm{13}} \\ $$$${n}=\pm\mathrm{10}^{\mathrm{9}} \:\Rightarrow\:{f}\left({x}\right)\approx−\mathrm{8}.\mathrm{9}×\mathrm{10}^{\mathrm{19}} \\ $$$$… \\ $$