Find the smallest and largest values of the function y = 5-2sin ^ 2 (2x) and its domain.

We have a function:

y = 5 – 2 * sin ^ 2 (2x).

To find the smallest and largest values of the function, it is worth noting that the trigonometric function of the sine of the argument, regardless of the value of its argument, takes values in the interval [-1; one]. Let’s write the range of values in the form of a double inequality:

-1 <= sin 2x <= 1;

We square the parts of the inequality:

0 <= sin ^ 2 2x <= 1;

Let’s multiply all the parts by minus two:

-2 <= -2 * sin ^ 2 2x <= 0;

Add five to all parts of the inequality:

3 <= 5 – 2 * sin ^ 2 2x <= 5.

3 and 5 – the smallest and largest values of the function.