Question

Dau coroana multe puncte este urgent va rog urgent!!!!!Plsssss

Answer 1

$5. \: \: \tt(u + 2v) {}^{2} \geq8uv \\\tt u {}^{2} + 4uv + 4v {}^{2} \geq8uv \\\tt u {}^{2} + 4v {}^{2} \geq4uv \\\tt u {}^{2} - 4uv + 4v {}^{2} \geq0 \\\tt u {}^{2} - 2\cdot2uv + (2v) {}^{2} \geq0 \\\tt (u - 2v) {}^{2} \geq0 \implies \: \text{adevarat} \\ \\ \\ 7. \: \sqrt{sin \: 3x - sin \: x} = \sqrt{2} sin \: x \\ sin \: 3x - sin \: x = 2sin {}^{2} \: x \\ - sin \: x - 2sin {}^{2} \: x + sin \: 3x = 0 \\ 2sin \: x - 2sin {}^{2} \: x - 4sin {}^{3} \: x = 0 \\ - 2sin \: x(sin \: x + 1)(2sin \: x - 1) = 0 \\ sin \: x(sin \: x + 1)(2sin \: x - 1) = 0 \\ \implies \: sin \: x = 0\implies \:\boxed{ x = \pi k} \\ \implies \: sin \: x + 1 = 0 \implies \:\boxed{ x = \frac{4\pi k - \pi}{2}} \\ \implies \: 2sin \: x - 1 = 0\implies \: \boxed{x = \frac{12\pi k - \pi}{6} } \\ unde \: k \: este \: intreg \\ \\ \text{dar singura solutie intreaga este} \boxed{x = 0}$

Sper ca te-am ajutat si cu atat!