Questions: Identify the absolute minimum and maximum values of the function on the given interval (when they exist). f(x)=4x-20tan^(-1)x ;[-3sqrt(3), 3sqrt(3)]

Solution Steps

To find the absolute minimum and maximum values of the function \( f(x) = 4x - 20 \tan^{-1} x \) on the interval \([-3\sqrt{3}, 3\sqrt{3}]\), we need to evaluate the function at critical points and endpoints. First, find the derivative of the function and set it to zero to find critical points. Then, evaluate the function at these critical points and at the endpoints of the interval. Compare these values to determine the absolute minimum and maximum.

We are given the function \( f(x) = 4x - 20 \tan^{-1} x \) and the interval \([-3\sqrt{3}, 3\sqrt{3}]\).

We calculate the function values at the endpoints:

• At \( x = -3\sqrt{3} \): \[ f(-3\sqrt{3}) \approx 6.8288 \]
• At \( x = 3\sqrt{3} \): \[ f(3\sqrt{3}) \approx -6.8288 \]

We find the critical points by minimizing the function. The minimum value occurs at:

• \( x \approx 1.999999 \) with \( f(1.999999) \approx -14.1430 \)

The maximum value occurs at:

• \( x \approx -1.999999 \) with \( f(-1.999999) \approx 14.1430 \)

Now we compare the function values at the endpoints and the critical points:

• \( f(-3\sqrt{3}) \approx 6.8288 \)
• \( f(3\sqrt{3}) \approx -6.8288 \)
• \( f(1.999999) \approx -14.1430 \)
• \( f(-1.999999) \approx 14.1430 \)

The absolute minimum value is: \[ \text{Absolute Minimum} \approx -14.1430 \] The absolute maximum value is: \[ \text{Absolute Maximum} \approx 14.1430 \]

Final Answer