Questions: Homework14: Problem 10 (1 point) Find all critical values for the function f(x) = 9 x^3 - 27 x + 5, and then list them (separated by commas) in the box below. List of critical numbers:

Solution Steps

To find the critical values of the function \( f(x) = 9x^3 - 27x + 5 \), we need to follow these steps:

1. Compute the first derivative of the function, \( f'(x) \).
2. Set the first derivative equal to zero and solve for \( x \) to find the critical points.
3. Verify the solutions by checking the second derivative or using the first derivative test.

To find the critical values of the function \( f(x) = 9x^3 - 27x + 5 \), we first compute the first derivative: \[ f'(x) = 27x^2 - 27 \]

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

We can simplify the equation: \[ 27(x^2 - 1) = 0 \implies x^2 - 1 = 0 \] Factoring gives: \[ (x - 1)(x + 1) = 0 \] Thus, the solutions are: \[ x = -1 \quad \text{and} \quad x = 1 \]

Final Answer