While solving some questions related to functions, I came across this question and I am unable to find its range.
$$f(x)=3\cos^4x-6\cos^3x-6\cos^2x-3$$in the interval $[-π/2, π/2]$
I tried graphing this function on Desmos and got this result.
However, since I can't use a graphing tool in exams, I need to be able to solve this algebraically.
When I tried factorising the function, I could only get this far:
$$f(x)=3(\cos x+1)(\cos^3x-3\cos^2x+\cos x-1)$$
Can anyone help how to solve this further?
Best Answer
Let $\varphi(x)=3x^4-6x^3-6x^2-3$; then $f(x)=\varphi\bigl(\cos(x)\bigr)$.
If $x\in[0,1]$, you have $\varphi'(x)\leqslant0$ and $\varphi'(x)=0\iff x=0$. So, $\varphi$ is decreasing on $[0,1]$. Since $\varphi(0)=-3$ and $\varphi(1)=-12$, the range of the restriction of $\varphi$ to $[0,1]$ is $[-12,-3]$ (see the graph of $\varphi$ below). Since the range of $\cos$ on $\left[-\frac\pi2,\frac\pi2\right]$ is $[0,1]$, the range of $f(=\varphi\circ\cos)$ is also $[-12,-3]$.