It is known that the domain of definition of the function f (x) = 2asin (2x - one third π) + B is r, the maximum value of the function is 2, and the minimum value is - 6

It is known that the domain of definition of the function f (x) = 2asin (2x - one third π) + B is r, the maximum value of the function is 2, and the minimum value is - 6

a>0
2a+b=2
-2a+b=-6
simultaneous
a=2,b=-2
a

Let f (x) = 2x + (1 / x) - 1 (x)

∵x0
-2x>0,-1/x>0
From mean inequality
-2x-(1/x)≥2√[(-2x)(-1/x)]=2√2
∴2x+(1/x)≤-2√2
f(x)=2x+(1/x)-1≤-2√2-1
That is, f (x) has a maximum value of - 2 √ 2-1
Conclusion: pay attention to the condition when using the mean inequality
When a > 0 and b > 0, (a + b) / 2 ≥ √ ab