Question Number 18402 by Tinkutara last updated on 20/Jul/17
![If P_n = cos^n θ + sin^n θ, θ ∈ [0, (π/2)], n ∈ (−∞, 2), then minimum of P_n will be (1) 1 (2) (1/2) (3) (√2) (4) (1/( (√2)))](https://www.tinkutara.com/question/Q18402.png)
$$\mathrm{If}\:{P}_{{n}} \:=\:\mathrm{cos}^{{n}} \:\theta\:+\:\mathrm{sin}^{{n}} \:\theta,\:\theta\:\in\:\left[\mathrm{0},\:\frac{\pi}{\mathrm{2}}\right],\:{n}\:\in \\ $$$$\left(−\infty,\:\mathrm{2}\right),\:\mathrm{then}\:\mathrm{minimum}\:\mathrm{of}\:{P}_{{n}} \:\mathrm{will}\:\mathrm{be} \\ $$$$\left(\mathrm{1}\right)\:\mathrm{1} \\ $$$$\left(\mathrm{2}\right)\:\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\left(\mathrm{3}\right)\:\sqrt{\mathrm{2}} \\ $$$$\left(\mathrm{4}\right)\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}} \\ $$
Answered by Tinkutara last updated on 28/Jul/17
$$\left(\mathrm{sin}\:\theta\right)^{−{n}} \:\geqslant\:…\:\geqslant\:\left(\mathrm{sin}\:\theta\right)^{−\mathrm{1}} \:\geqslant\:\mathrm{sin}\:\theta\:\geqslant\:\mathrm{sin}^{\mathrm{2}} \:\theta \\ $$$$\left(\mathrm{cos}\:\theta\right)^{−{n}} \:\geqslant\:…\:\geqslant\:\left(\mathrm{cos}\:\theta\right)^{−\mathrm{1}} \:\geqslant\:\mathrm{cos}\:\theta\:\geqslant\:\mathrm{cos}^{\mathrm{2}} \:\theta \\ $$$$\left(\mathrm{sin}\:\theta\right)^{−{n}} \:+\:\left(\mathrm{cos}\:\theta\right)^{−{n}} \:>\:\mathrm{sin}^{\mathrm{2}} \:\theta\:+\:\mathrm{cos}^{\mathrm{2}} \:\theta\:=\:\mathrm{1} \\ $$$$\therefore\:\mathrm{Minimum}\:\mathrm{of}\:{P}_{{n}} \:=\:\mathrm{1} \\ $$