Find the min of 22

$\begin{aligned} y & = 3-4\sin^2 x + \cos^2 x \\ & = 3 - 4\sin^2 x + 1 - \sin^2 x \\ & = 4 - 5\blue{\sin^2 x} & \small \blue{\text{Note that }0 \le \sin^2 x \le 1} \\ \implies y_{\min} & = \boxed{-1} & \small \blue{\text{when }\sin^2 x = 1} \\ y_{\max} & = 4 & \small \blue{\text{when }\sin^2 x = 0} \end{aligned}$

Therefore $y \in [-1, 4]$ .

$y=3-4\sin^2 x+\cos^2 x=4-5\sin^2 x\geq -1$ (since $\sin^2 x\leq 1$ )

So, the minimum of $y$ is $\boxed {-1}$ and it's range is $\boxed {[-1,4]}$ (since $\sin^2 x\geq 0$ ) .

To find the minimum of y, you first need to find the derivative of dy/dx to find at which x value dy/dx becomes zero. The derivative ends up being:

$y’=-10cos(x)sin(x)$

This means that $y’$ can be zero at $x=-1$ and $x=0$ , meaning that the minimum is found at $x=-1$ .