how to get the maximum value of function f:f(X) = 2 sin 4X
To find the maximum value of the function f(X) = 2sin(4X), you need to find the maximum value of sin(4X).
The maximum value of the sine function is 1. This occurs at certain values of X.
In this case, to find the maximum value of sin(4X), you need to find the values of X that make 4X equal to the specific angles where sin(x) is equal to 1.
The general form for such angles is X = (2n + 1)π/2, where n is an integer.
So, to find the maximum value of sin(4X), you need to solve the equation:
4X = (2n + 1)π/2
Solving for X, you get:
X = (2n + 1)π/8
Now, to find the maximum value of f(X) = 2sin(4X), substitute these values of X into the function and compare them to find the maximum value.
f((2n + 1)π/8) = 2sin(4((2n + 1)π/8)) = 2sin((2n + 1)π/2)
Since sin((2n + 1)π/2) = 1, the maximum value of f(X) = 2sin(4X) is:
f((2n + 1)π/8) = 2(1) = 2
Therefore, the maximum value of the function f(X) = 2sin(4X) is equal to 2.