Find the Range of \( f(x)=\cos^2x+\sin^4x \)
The function
\[ f(x)=\cos^2x+\sin^4x \]
is defined from \(R\to R\). Then,
\[ f(R)=? \]
(a) \(\left[\frac34,1\right)\)
(b) \(\left(\frac34,1\right]\)
(c) \(\left[\frac34,1\right]\)
(d) \(\left(\frac34,1\right)\)
Using
\[ \cos^2x=1-\sin^2x \]
\[ f(x)=1-\sin^2x+\sin^4x \]
Let
\[ \sin^2x=t \]
where
\[ 0\le t\le1 \]
Then,
\[ f(x)=t^2-t+1 \]
\[ =\left(t-\frac12\right)^2+\frac34 \]
Minimum value:
\[ \frac34 \]
Maximum value:
\[ 1 \]
Therefore,
\[ f(R)=\left[\frac34,1\right] \]
\[ \boxed{\text{Correct Answer: (c)}} \]