Questions: Determine the location and value of the absolute extreme values of f on the given interval, if they exist. f(x)=-5 sec x ;[-π/3, π/3] Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. (Type exact answers, using π as needed. Use a comma to separate answers as needed.) A. The absolute maximum is at x= , but there is no absolute minimum. B. The absolute minimum is at x= , but there is no absolute maximum. C. The absolute minimum is at x= and the absolute maximum is at x= . D. There are no absolute extreme values for f(x) on [-π/3, π/3].

Solution Steps

We evaluate the function \( f(x) = -5 \sec x \) at the endpoints of the interval \( x = -\frac{\pi}{3} \) and \( x = \frac{\pi}{3} \).

Calculating the values: \[ f\left(-\frac{\pi}{3}\right) = -5 \sec\left(-\frac{\pi}{3}\right) = -5 \cdot -2 = -10 \] \[ f\left(\frac{\pi}{3}\right) = -5 \sec\left(\frac{\pi}{3}\right) = -5 \cdot -2 = -10 \]

Both evaluations yield the same value: \[ f\left(-\frac{\pi}{3}\right) = f\left(\frac{\pi}{3}\right) = -10 \] Thus, the absolute maximum and minimum values are both \(-10\).

The corresponding \( x \) values for the absolute maximum and minimum are: \[ x = -\frac{\pi}{3} \quad \text{and} \quad x = \frac{\pi}{3} \]

Since both the absolute maximum and minimum occur at the endpoints, we conclude that: \[ \text{Absolute maximum: } -10 \text{ at } x = -\frac{\pi}{3} \text{ and } x = \frac{\pi}{3} \] \[ \text{Absolute minimum: } -10 \text{ at } x = -\frac{\pi}{3} \text{ and } x = \frac{\pi}{3} \]

Final Answer