Determine the minimum and maximum value of the following trigonometric function.

Question

f, of, x, equals, 4, cosine, 3, x, plus, 3
f(x)=4cos3x+3

Answer 1

To find the minimum and maximum values of the function \( f(x) = 4 \cos(3x) + 3 \), we first analyze the cosine component.

1. Understand the cosine function: The function \( \cos(3x) \) oscillates between -1 and 1 for all values of \( x \).

2. Apply this to the function: \[ f(x) = 4 \cos(3x) + 3 \] Let's find the extreme values of \( f(x) \) by evaluating the extreme values of \( \cos(3x) \):

   • The maximum value of \( \cos(3x) \) is 1.
   • The minimum value of \( \cos(3x) \) is -1.

3. Calculate the max and min of \( f(x) \):

   • Maximum value: \[ f_{\text{max}} = 4(1) + 3 = 4 + 3 = 7 \]

   • Minimum value: \[ f_{\text{min}} = 4(-1) + 3 = -4 + 3 = -1 \]

4. Conclusion: Thus, the minimum and maximum values of the function \( f(x) = 4 \cos(3x) + 3 \) are:

   • Minimum value: \( -1 \)
   • Maximum value: \( 7 \)