I am working on my scholarship exam practice which assumes high school or pre-university math knowledge. Could you please have a look on my approach?
The minimum of the function $f(x)=(2+\sin x)(5-\sin x)$ is ……
First, I began with some basic approach.
We know that $-1\leq\sin x\leq1$, so I just test both values $-1$ and $1$ in the function.
$f(-1)=(2+(-1))(5-(-1))=6$
$f(1)=(2+1)(5-1)=12$
Since $-1$ is the minimum of function $\sin x$, I conclude that $f(-1)=6$, which is correct when checked with the value of answer key provided. Please let me know if my approach is not always true or can apply on other similar problems.
But in exam I may be uncertain if my answer is right so I tried calculus approach to check my answer.
$f(x)=(2+\sin x)(5-\sin x)=10+3\sin x-\sin^2 x$
$f'(x)=3\cos x-2\sin x\cos x=0$
$2\sin x\cos x=3\cos x$
$2\sin x=3$
$\sin x=\frac{3}{2}=1.5>1$
Since 1.5 exceeds the range of $\sin x$, then I cannot use this approach. I am wondering why this is the case. Why can't we use this method to find the minimum?
Best Answer
We have $2 \sin x \cos x=3\ cos x \iff 2 \sin x =3$ or $ \cos x=0.$