Final answer:
The function f(x) = 2x^3 - 6x + 7 has a local minimum at x = 1 and a local maximum at x = -1, determined by setting the first derivative to zero and using the second derivative test.
Step-by-step explanation:
The question asks to determine the stationary values and their types for the function f(x) = 2x^3 - 6x + 7. To find the stationary values, we need to calculate the function's derivative and set it equal to zero, then solve for x. The types of stationary points can be determined by the second derivative test.
Step-by-step process:
- Find the first derivative: f'(x) = 6x^2 - 6.
- Set the first derivative equal to zero to find critical points: 0 = 6x^2 - 6.
- Solve for x: x = ± 1.
- Find the second derivative: f''(x) = 12x.
- Determine the type of stationary points by plugging the values of x into f''(x):
- For x = 1, f''(1) = 12 (positive, indicating a local minimum).
- For x = -1, f''(-1) = -12 (negative, indicating a local maximum).
Therefore, the function f(x) = 2x^3 - 6x + 7 has a local minimum at x = 1 and a local maximum at x = -1.