Question

find the extreme values of the following functions over R
4 sin^2 x + 5 cos^2 x + 6
​

Answer 1

$f(x) = 4 \sin^{2} (x) + 5 { \cos}^{2} (x) + 6$

$∀ \: x \in ℝ$

$f(x) = 4 { \sin}^{2} (x) + 11 - 5 \sin ^{2} (x)$

$f(x) = 11 - { \sin}^{2} (x)$

$0 \leq\sin ^{2} (x) \leq1$

$0 \geq - { \sin}^{2} (x) \geq - 1$

$11 \geq11 - { \sin}^{2} (x) \geq10$

$\implies \boxed{f(x) \in[10,11]}$