Find the Largest and Smallest Values of the Function y = 1 / 3cos ^ 2x-1 / 3sin ^ 2x + 1

We have a function:

y = 1/3 * cos ^ 2 x – 1/3 * sin ^ 2 x + 1.

We transform the function formula using the double angle cosine formula:

y = 1/3 * (cos ^ 2 x – sin ^ 2 x) + 1;

y = 1/3 * cos 2x + 1.

The cosine of any argument will range from minus one to one. We write the range of values of the function in the form of a double inequality:

-1 <= cos 2x <= 1;

Multiply all parts of the inequality by 1/3:

-1/3 <= 1/3 * cos 2x <= 1/3;

Let’s add one:

2/3 <= 1/3 * cos 2x + 1 <= 4/3.

2/3 and 4/3 are the smallest and largest values of the function, respectively.