<s>[INST] minimum value of $2\cos \alpha\sin \beta+3\sin \alpha\sin \beta+4\cos \beta$ Let $\alpha,\beta$ be real numbers ; find the minimum value of 
$2\cos \alpha\sin \beta+3\sin \alpha\sin \beta+4\cos \beta$
What I tried :
$\bigg|4\cos \beta+(2\cos \alpha+3\sin \alpha)\sin \beta\bigg|^2 \leq 4^2+(2\cos \alpha+3\sin \alpha)^2$
How do I solve it ? Help me please
 [/INST] By C-S twice we obtain: 
$$2\cos\alpha\sin\beta+3\sin\alpha\sin\beta+4\cos\beta=$$
$$=\sin\beta(2\cos\alpha+3\sin\alpha)+4\cos\beta\geq$$
$$\geq-\sqrt{(\sin^2\beta+\cos^2\beta)((2\cos\alpha+3\sin\alpha)^2+16)}\geq$$
$$\geq-\sqrt{\left(\sqrt{2^2+3^2)(\cos^2\alpha+\sin^2\alpha}\right)^2+16}=-\sqrt{29}.$$
The equality occurs for $(\cos\alpha,\sin\alpha)||(2,3)$ and $(\sin\beta,\cos\beta)||(2\cos\alpha+3\sin\alpha,4),$ 
which says that we got a minimal value.
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