how to get the maximum value of function f:f(x) = 2sin4x

Answer 1

To find the maximum value of the function f(x) = 2sin(4x), we can start by finding the critical points.

First, let's find the derivative of f(x) with respect to x:
f'(x) = 8cos(4x)

Now, set f'(x) equal to zero and solve for x:
8cos(4x) = 0

To find the critical points, we need to find the values of x that satisfy this equation.

cos(4x) = 0 when 4x = π/2 + nπ/2, where n is an integer.

Solving for x, we have:
x = (π/2 + nπ/2) / 4, where n is an integer.

Now we can evaluate f(x) at these critical points and find the maximum value of the function.

f(x) = 2sin(4x)

Let's substitute the values of x we found earlier into f(x):

When x = (π/2 + nπ/2) / 4:
f(x) = 2sin(4(π/2 + nπ/2) / 4)
= 2sin(π/2 + nπ/2)
= 2sin(π/2)
= 2(1)
= 2

Therefore, the maximum value of the function f(x) = 2sin(4x) is 2.