e total number of local maxima and local minima of the function `f(x) = {(2+x)^3, -3

Options

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

Answer

To find the total number of local maxima and local minima of the function

\[ f(x) = \begin{cases} (2+x)^3 & \text{for } -3 < x \leq -1 \\ x^3 & \text{for } -1 < x < 2 \end{cases} \]

we will follow these steps:

### Step 1: Differentiate the function
We need to find the derivative of the function \( f(x) \) in both intervals.

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

2. For the second piece \( f(x) = x^3 \):
\[ f'(x) = 3x^2 \]

### Step 2: Find critical points
Next, we will set the derivatives equal to zero to find critical points.

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

2. For \( -1 < x < 2 \):
\[ 3x^2 = 0 \implies x^2 = 0 \implies x = 0 \]
This point \( x = 0 \) is within the interval \( -1 < x < 2 \).

### Step 3: Analyze the sign of the derivative
Now, we will analyze the sign of the derivative \( f'(x) \) around the critical points to determine if they are local maxima or minima.

1. **For \( x = -2 \)**:
- Choose a test point in \( (-3, -2) \), say \( x = -2.5 \):
\[ f'(-2.5) = 3(2 - 2.5)^2 = 3(-0.5)^2 > 0 \quad (\text{increasing}) \]
- Choose a test point in \( (-2, -1) \), say \( x = -1.5 \):
\[ f'(-1.5) = 3(2 - 1.5)^2 = 3(0.5)^2 > 0 \quad (\text{increasing}) \]
- Since \( f' \) does not change sign around \( x = -2 \), it is neither a local maximum nor a local minimum.

2. **For \( x = 0 \)**:
- Choose a test point in \( (-1, 0) \), say \( x = -0.5 \):
\[ f'(-0.5) = 3(-0.5)^2 > 0 \quad (\text{increasing}) \]
- Choose a test point in \( (0, 2) \), say \( x = 1 \):
\[ f'(1) = 3(1)^2 > 0 \quad (\text{increasing}) \]
- Since \( f' \) does not change sign around \( x = 0 \), it is neither a local maximum nor a local minimum.

### Conclusion
Since both critical points \( x = -2 \) and \( x = 0 \) do not correspond to local maxima or minima, we conclude that the function \( f(x) \) has:

\[ \text{Total number of local maxima and local minima} = 0 \]