The function `f(x) = 2x + 3^(2/3)`, `x in`, has

Options

A. exactly one point of local minima and no
point of local maxima
B. exactly one point of local maxima and no
point of local minima
C. exactly one point of local maxima and exactly one point of local minima
D. exactly two points of local maxima and exactly one point of local minima

Answer

To solve the problem regarding the function \( f(x) = 2x + 3^{2/3} \), we will analyze its behavior by finding its critical points and determining the nature of these points.

### Step-by-Step Solution:

1. **Identify the Function**:
\[ f(x) = 2x + 3^{2/3} \]
Here, \( 3^{2/3} \) is a constant.

2. **Find the Derivative**:
To find the critical points, we first need to compute the derivative \( f'(x) \).
\[ f'(x) = \frac{d}{dx}(2x + 3^{2/3}) = 2 \]
Since \( 3^{2/3} \) is a constant, its derivative is 0.

3. **Set the Derivative to Zero**:
We set the derivative equal to zero to find critical points:
\[ 2 = 0 \]
This equation has no solution, indicating that there are no critical points where the derivative is zero.

4. **Analyze the Behavior of the Function**:
Since \( f'(x) = 2 \) is a constant and positive, this means that the function \( f(x) \) is strictly increasing for all \( x \).

5. **Determine the Nature of the Function**:
- Since the function is strictly increasing, it does not have any local maxima or minima.
- The function approaches \( -\infty \) as \( x \) approaches \( -\infty \) and \( +\infty \) as \( x \) approaches \( +\infty \).

6. **Conclusion**:
Since there are no critical points and the function is strictly increasing, we conclude that:
- The function has no maximum or minimum values.

### Final Answer:
The function \( f(x) = 2x + 3^{2/3} \) has no maximum or minimum.