The total number of local maxima and local minima of the function f(x) = `{{:( (2 + x)^(3) "," , -3 lt x le -1) , (x^(2//3) "," , -1 lt x lt 2):}` is

Options

A. 0
B. 1
C. 2
D. 3

Answer

To find the total number of local maxima and local minima of the function given, we will analyze the function piecewise and find the critical points by differentiating the function.

### Step-by-Step Solution:

1. **Define the Function:**
The function is defined piecewise as follows:
\[ f(x) = \begin{cases} (2 + x)^3 & \text{for } -3 < x \leq -1 \\ x^{2/3} & \text{for } -1 < x < 2 \end{cases} \]

2. **Differentiate Each Piece:**
We need to find the derivative \( f'(x) \) for each piece of the function.

- For the first piece \( f(x) = (2 + x)^3 \):
\[ f'(x) = 3(2 + x)^2 \cdot (1) = 3(2 + x)^2 \]

- For the second piece \( f(x) = x^{2/3} \):
\[ f'(x) = \frac{2}{3} x^{-1/3} = \frac{2}{3\sqrt[3]{x^2}} \]

3. **Find Critical Points:**
Set the derivatives equal to zero to find critical points.

- For \( f'(x) = 3(2 + x)^2 \):
\[ 3(2 + x)^2 = 0 \implies (2 + x)^2 = 0 \implies x = -2 \]
This critical point \( x = -2 \) is in the interval \( -3 < x \leq -1 \).

- For \( f'(x) = \frac{2}{3\sqrt[3]{x^2}} \):
The derivative does not equal zero for any \( x \) in the interval \( -1 < x < 2 \) since \( x^{2/3} \) is always positive in this interval.

4. **Check Endpoints:**
We also need to check the endpoints of the intervals:
- At \( x = -1 \):
\[ f(-1) = (-1)^{2/3} = 1 \]
- At \( x = 2 \):
\[ f(2) = 2^{2/3} \text{ (which is positive)} \]

5. **Determine Local Maxima and Minima:**
- The critical point \( x = -2 \) is the only critical point found in the first interval.
- Since \( f'(x) > 0 \) for \( x > -2 \) in the first piece, it indicates that \( x = -2 \) is a local minimum.
- In the second piece, there are no critical points, and since the function is increasing, there are no local maxima or minima.

6. **Count Local Maxima and Minima:**
- From the analysis, we have:
- 1 local minimum at \( x = -2 \)
- 0 local maxima

### Conclusion:
The total number of local maxima and local minima of the function is:
\[ \text{Total = 1 (local minimum) + 0 (local maxima) = 1} \]