Find the maximum or minimum values, if any, of the following funcitons without using the derivatives :
`g(x)=x^(3)+1`.

Answer

To find the maximum or minimum values of the function \( g(x) = x^3 + 1 \) without using derivatives, we can analyze the function graphically and logically.

### Step-by-Step Solution:

1. **Understanding the Function**:
The function \( g(x) = x^3 + 1 \) is a cubic function. Cubic functions are continuous and can take on all real values as \( x \) approaches positive or negative infinity.

2. **Identifying Key Points**:
Let's evaluate the function at some key points:
- For \( x = 0 \):
\[ g(0) = 0^3 + 1 = 1 \]
- For \( x = 1 \):
\[ g(1) = 1^3 + 1 = 2 \]
- For \( x = -1 \):
\[ g(-1) = (-1)^3 + 1 = 0 \]
- For \( x = 2 \):
\[ g(2) = 2^3 + 1 = 9 \]
- For \( x = -2 \):
\[ g(-2) = (-2)^3 + 1 = -7 \]

3. **Graphing the Function**:
By plotting these points, we can visualize the function:
- The point \( (0, 1) \) is on the graph.
- The point \( (1, 2) \) shows that the function is increasing.
- The point \( (-1, 0) \) indicates that the function is also increasing from negative values.
- The function continues to increase as \( x \) increases and decreases as \( x \) decreases.

4. **Behavior at Infinity**:
As \( x \) approaches positive infinity (\( x \to +\infty \)), \( g(x) \) also approaches positive infinity. Similarly, as \( x \) approaches negative infinity (\( x \to -\infty \)), \( g(x) \) approaches negative infinity.

5. **Conclusion**:
Since the function \( g(x) = x^3 + 1 \) is continuous and increases without bound in both directions, it does not have a maximum or minimum value. The function can take any real number as its output.

### Final Answer:
The function \( g(x) = x^3 + 1 \) has no maximum or minimum values.

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