Let `f(x)={{:((x+1)^(3),-2ltxle-1),(x^(2//3)-1,-1ltxle1),(-(x-1)^(2),1ltxlt2):}`. The total number of maxima and minima of f(x) is

Options

A. 4
B. 3
C. 2
D. 1

Explanation

To find the total number of maxima and minima of the piecewise function \( f(x) \), we will follow these steps:

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

### Step 2: Find the Derivative
We need to find the derivative \( f'(x) \) for each piece of the function.

1. For \( -2 < x \leq -1 \):
\[ f'(x) = 3(x + 1)^2 \]

2. For \( -1 < x \leq 1 \):
\[ f'(x) = \frac{2}{3}x^{-1/3} \]

3. For \( 1 < x < 2 \):
\[ f'(x) = -2(x - 1) \]

### Step 3: Find Critical Points
Set the derivative equal to zero to find critical points:

1. For \( -2 < x \leq -1 \):
\[ 3(x + 1)^2 = 0 \implies x + 1 = 0 \implies x = -1 \]
(This point is at the boundary and will be checked later.)

2. For \( -1 < x \leq 1 \):
\[ \frac{2}{3}x^{-1/3} = 0 \implies \text{No solution in this interval.} \]

3. For \( 1 < x < 2 \):
\[ -2(x - 1) = 0 \implies x - 1 = 0 \implies x = 1 \]
(This point is also at the boundary.)

### Step 4: Check the Boundary Points
We need to check the behavior of the function at the boundaries and critical points:

- At \( x = -1 \):
- \( f(-1) = (-1 + 1)^3 = 0 \)

- At \( x = 1 \):
- \( f(1) = 1^{2/3} - 1 = 0 \)

### Step 5: Analyze the Intervals
Now, we will analyze the sign of \( f'(x) \) in the intervals:

1. For \( -2 < x < -1 \):
- \( f'(x) > 0 \) (increasing)

2. For \( -1 < x < 1 \):
- \( f'(x) > 0 \) (increasing)

3. For \( 1 < x < 2 \):
- \( f'(x) < 0 \) (decreasing)

### Step 6: Determine Maxima and Minima
From the analysis:
- At \( x = -1 \): \( f(x) \) is increasing before and at this point, so it is not a maximum or minimum.
- At \( x = 1 \): \( f(x) \) is increasing before and decreasing after, indicating a local maximum.

### Conclusion
The total number of maxima and minima of the function \( f(x) \) is:
- **1 maximum** at \( x = 1 \)
- **0 minima**

Thus, the total number of maxima and minima of \( f(x) \) is **1**.

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