Questions: Find the absolute extremum, if any, for the following function. f(x)=9x^3-1 Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. The absolute minimum is at x=. B. There is no absolute minimum.

Solution Steps

To find the absolute extremum of the function \( f(x) = 9x^3 - 1 \), we need to follow these steps:

1. Find the first derivative of the function.
2. Set the first derivative equal to zero to find critical points.
3. Evaluate the function at the critical points and endpoints (if any) to determine the absolute extremum.

1. Compute the first derivative of \( f(x) \).
2. Solve \( f'(x) = 0 \) to find critical points.
3. Evaluate \( f(x) \) at the critical points to find the absolute extremum.

The function is given by \( f(x) = 9x^3 - 1 \). To find the critical points, we first compute the first derivative: \[ f'(x) = 27x^2 \]

Next, we set the first derivative equal to zero to find the critical points: \[ 27x^2 = 0 \] This gives us the critical point: \[ x = 0 \]

Now, we evaluate the original function at the critical point \( x = 0 \): \[ f(0) = 9(0)^3 - 1 = -1 \]

Final Answer