Find the largest and smallest value of the function y = 2cos3x-4.

1. If the argument of the function cosx takes all possible values, then the range of values of the function is the interval [-1; one]:

cos (3x) ∈ [-1; 1], or
-1 ≤ cos (3x) ≤ 1. (1)
2. Multiply all parts of the double inequality (1) by 2, then subtract 4:

-2 ≤ 2cos (3x) ≤ 2;
-2 – 4 ≤ 2cos (3x) – 4 ≤ 2 – 4;
-6 ≤ 2cos (3x) – 4 ≤ -2;
-6 ≤ y ≤ -2;
y ∈ [-6; -2].
3. The function y = 2cos (3x) – 4 takes values in the interval [-6; -2], hence the smallest and largest values:

y (min) = -6;
y (max) = -2.
Answer:

smallest value: -6;
highest value: -2.