finding $k$ if condition is given

Question

My attempt:

Thoughtfully can it be said that the min value of this function $ = -\dfrac 7k$. Am I wrong in saying this. Please help me find out.

Answer 1

The minimum of $\cos^2x\sin x$ is $m=-2\sqrt{3}/9$. Then we must find integer $k$ such that:

$$-{7\over k}<m<-{7\over k+1},$$

that is:

$$ k<-{7\over m}<k+1. $$ But $-7/m=21\sqrt3/2\approx18.19$, hence $k=18$.