`f(x) = 2^(x) + 1` and `g(x) = x^(3) - (x)/(2) + 2`.At which of the points given by the options below do the graphs of the two function intersect ?
I. (x,y) = (-1,2)
II. (x,y) = (0,2)
III (x,y) = (1,3) .

Options

A. Only I
B. Only II
C. Only III
D. Both I and II

Answer

To find the points of intersection of the functions \( f(x) = 2^x + 1 \) and \( g(x) = x^3 - \frac{x}{2} + 2 \), we will evaluate both functions at the given points and check where they are equal.

### Step 1: Evaluate at Point I: \((-1, 2)\)

1. **Calculate \( f(-1) \)**:
\[ f(-1) = 2^{-1} + 1 = \frac{1}{2} + 1 = \frac{3}{2} \]

2. **Calculate \( g(-1) \)**:
\[ g(-1) = (-1)^3 - \frac{-1}{2} + 2 = -1 + \frac{1}{2} + 2 = -1 + 0.5 + 2 = 1.5 \]

3. **Comparison**:
\[ f(-1) = \frac{3}{2} \quad \text{and} \quad g(-1) = 1.5 \]
Since \( f(-1) \neq g(-1) \), the graphs do not intersect at this point.

### Step 2: Evaluate at Point II: \((0, 2)\)

1. **Calculate \( f(0) \)**:
\[ f(0) = 2^0 + 1 = 1 + 1 = 2 \]

2. **Calculate \( g(0) \)**:
\[ g(0) = 0^3 - \frac{0}{2} + 2 = 0 - 0 + 2 = 2 \]

3. **Comparison**:
\[ f(0) = 2 \quad \text{and} \quad g(0) = 2 \]
Since \( f(0) = g(0) \), the graphs intersect at this point.

### Step 3: Evaluate at Point III: \((1, 3)\)

1. **Calculate \( f(1) \)**:
\[ f(1) = 2^1 + 1 = 2 + 1 = 3 \]

2. **Calculate \( g(1) \)**:
\[ g(1) = 1^3 - \frac{1}{2} + 2 = 1 - 0.5 + 2 = 1.5 + 2 = 2.5 \]

3. **Comparison**:
\[ f(1) = 3 \quad \text{and} \quad g(1) = 2.5 \]
Since \( f(1) \neq g(1) \), the graphs do not intersect at this point.

### Conclusion
The graphs of the functions intersect only at point II: \((0, 2)\).