If A=4sinΘ+cos2Θ,then which of the following is true?
Consider
f(x)=4sinx+cos2x
f(x)=4sinx+1−sin2(x)
Therefore
f(x)=−sin2(x)+4sin(x)+1
Now
sin(x)ϵ[−1,1]
Thus the function must attain a minimum value at sin(x)=−1.
That is , at x=−π2,3π2
Hence
f(−π2)=f(3π2)=−1+4(−1)+1
=−4
Hence minimum value of the function is −4.
Similarly, it must attain a maximum value at x=π2.
Therefore
f(π2)=−1+4+1=4
Thus the range of f(x) is [−4,4].