Functions#13

For $f(x) = \cos^4 x + \sin^2 x$ .

$\begin{aligned} f(x) & = \cos^4 x + \sin^2 x \\ & = (1-\sin^2x)^2+\sin^2 x \\ & = \sin^4 x - 2\sin^2 x + 1 + \sin^2 x \\ & = \sin^4 x - \sin^2 x + 1 \\ & = \left(\sin^2 x - \frac 12\right)^2 + \frac 34 \\ & = \left(\frac 12(1-\cos 2x) - \frac 12\right)^2 + \frac 34 \\ & = \frac 14 \cos^2 2x + \frac 34 \\ & = \frac 18 (1 + \cos 4x) + \frac 34 \\ & = \frac 18 \cos 4x + \frac 78 \end{aligned}$

Therefore,

1. The period of $f(x)$ is $4x = 2\pi \implies x = \color{#3D99F6}\frac \pi 2$ .
2. The range of $f(x)$ : $- \frac 18 + \frac 78 \le f(x) \le \frac 18 + \frac 78$ $\implies \frac 34 \le f(x) \le 1$ . There is only one integer 1 in the range. The statement "There are exactly 2 integers in the range of $f(x)$ " is $\color{#D61F06}\boxed{\text{false}}$ .
3. Since $\cos 4x$ is an even function $f(x)$ is also an $\color{#3D99F6}\text{even}$ function.
4. $\cos x$ is symmetrical about $x=\pi$ , therefore, $\cos 4x$ and hence $f(x)$ is symmetrical about $\color{#3D99F6} x=\frac \pi 4$ .

The range of the function is $[\frac34, 1]$ .

All other statements are true.