To find the coordinates of the turning point for the function f(x) = x³ + 3x + 1, we first need to calculate the derivative of the function. The turning points occur where the derivative is equal to zero, as these are the locations where the function may change direction (i.e., from increasing to decreasing or vice versa).
1. **Calculate the Derivative**: The first step is to differentiate the function.
f'(x) = d/dx (x³ + 3x + 1) = 3x² + 3
2. **Set the Derivative to Zero**: We then set the derivative equal to zero to find the x-coordinates of the turning points:
3x² + 3 = 0
Simplifying this gives:
x² + 1 = 0.
This leads to:
x² = -1.
Since there are no real solutions to this equation, it reveals that the function does not have any turning points on the real number line. Consequently, we can conclude that:
3. **Conclusion**: The function f(x) = x³ + 3x + 1 does not have any real turning points. Given the nature of cubic functions, it continues to increase or decrease without reversing direction in the real number domain.
So, the final answer is that there are no turning points for this function, and therefore, no coordinates to provide.