Question

Determine K de modo que se verifique a igualdade de:

Answer 1

Sabemos que a função cosseno está limitada no intervalo de [-1,1], portanto :

$\displaystyle \sf -1\leq cos\ x \leq 1 \\\\ -1\leq \frac{4k+1}{2}\leq 1 \\\\\\ \frac{4k+1}{2}\leq 1 \to 4k+1 \leq 2 \to 4k \leq 1 \to \boxed{\sf k \leq \frac{1}{4} } \\\\\\ \frac{4k+1}{2} \geq -1 \to 4k+1\geq -2 \to 4k\geq -3\to \boxed{\sf k \geq \frac{-3}{4} }\\\\\\ Portanto :\\\\ \huge\boxed{\sf \ \frac{-3}{4}\leq k \leq \frac{1}{4} \ }\checkmark$