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

f, of, x, equals, 4, sine, 4, x, minus, 1
f(x)=4sin4x−1

1 answer

To determine the minimum and maximum values of the function \( f(x) = 4\sin(4x) - 1 \), we need to analyze the behavior of the sine function.

  1. Understanding the sine component: The sine function \( \sin(4x) \) oscillates between \(-1\) and \(1\).

  2. Finding the maximum and minimum of \(4\sin(4x)\):

    • The maximum value of \( \sin(4x) \) is \(1\).
    • The minimum value of \( \sin(4x) \) is \(-1\).
    • Therefore: \[ \text{Maximum of } 4\sin(4x) = 4 \cdot 1 = 4 \] \[ \text{Minimum of } 4\sin(4x) = 4 \cdot (-1) = -4 \]
  3. Adjusting for the constant term \(-1\):

    • To find the corresponding maximum value of \( f(x) \): \[ f_{\text{max}} = 4 - 1 = 3 \]
    • To find the corresponding minimum value of \( f(x) \): \[ f_{\text{min}} = -4 - 1 = -5 \]
  4. Conclusion:

    • The maximum value of \( f(x) \) is \(3\).
    • The minimum value of \( f(x) \) is \(-5\).

Thus, the final result is: \[ \text{Maximum value: } 3, \quad \text{Minimum value: } -5 \]