Question

If $A=4\sin \Theta +{\cos }^2\Theta$,then which of the following is true?

• A. maximum value of $A$ is $5$.
• B. minimum value of $A$ is $-4$
• C. maximum value of $A$ occurs when $\sin \Theta =\frac{1}{2}$
• D. minimum value of $A$ occurs when $\sin \Theta =1$

Answer 1

The correct option is B minimum value of $A$ is $-4$

Consider

$f\left(x\right)=4\sin x+{\cos }^{2}x$

$f\left(x\right)=4\sin x+1-{\sin }^2\left(x\right)$
Therefore

$f\left(x\right)=-{\sin }^2\left(x\right)+4\sin \left(x\right)+1$

Now

$\sin \left(x\right)ϵ[-1,1]$

Thus the function must attain a minimum value at $\sin \left(x\right)=-1$.

That is , at $x=\frac{-\pi }{2},\frac{3\pi }{2}$

Hence

$f\left(\frac{-\pi }{2}\right)=f\left(\frac{3\pi }{2}\right)=-1+4(-1)+1$
$=-4$
Hence minimum value of the function is $-4.$

Similarly, it must attain a maximum value at $x=\frac{\pi }{2}$.

Therefore

$f\left(\frac{\pi }{2}\right)=-1+4+1=4$

Thus the range of $f\left(x\right)$ is $[-4,4]$.