Question

Cum gasesc minimul expresiei cos^2x-4sinx?

Answer 1

Am atasat rezolvarea.

36057144b8928793890f7d1824f0f2c0.pdf

Answer 2

$cos^2x-4sinx = 1-sin^2 x-4sinx = -sin^2x-4sinx+1 \\ \\ \boxed{1}\quad -1\leq sinx \leq 1 \Big|^2 \Rightarrow sin^2x \leq1 \Big|\cdot(-1) \Rightarrow \boxed{-sin^2x \geq -1} \\ \\ \boxed{2} \quad -1\leq sinx \leq 1 \Big|\cdot(-4) \Rightarrow 4 \geq -4sinx \geq -4 \Rightarrow \boxed{-4sinx \geq -4}\\ \\ Adunam cele 2 inegalitati: \Rightarrow -sin^2 x -4sinx \geq -1-4 \R \Rightarrow \\ \\ \Rightarrow -sin^2 x-4sinx\geq -5 \Big|+1 \Rightarrow -sin^2 x-4sinx+1 \geq -5+1 \Rightarrow$

$\Rightarrow -sin^2 x-4sinx+1 \geq -4 \Leftrightarrow cos^2x -4sinx \geq -4 \\ \\ \rightarrow minimul expresiei este -4$