Let f(x) = `{{:( 2 sin x ,, x gt 0) , ( x^(2) ,, x le 0):}` . Investigate the function for maxima/minima at x = 0 .

Answer

To investigate the function \( f(x) \) for maxima and minima at \( x = 0 \), we will follow these steps:

### Step 1: Define the function
The function is given as:
\[ f(x) = \begin{cases} 2 \sin x & \text{for } x > 0 \\ x^2 & \text{for } x \leq 0 \end{cases} \]

### Step 2: Find the first derivative \( f'(x) \)
We will differentiate \( f(x) \) for both cases:

- For \( x > 0 \):
\[ f'(x) = 2 \cos x \]

- For \( x \leq 0 \):
\[ f'(x) = 2x \]

### Step 3: Evaluate the first derivative at \( x = 0 \)
We need to check the derivative at \( x = 0 \):
\[ f'(0) = 2(0) = 0 \]

### Step 4: Find the second derivative \( f''(x) \)
Now we will differentiate \( f'(x) \):

- For \( x > 0 \):
\[ f''(x) = -2 \sin x \]

- For \( x \leq 0 \):
\[ f''(x) = 2 \]

### Step 5: Evaluate the second derivative at \( x = 0 \)
Next, we check the second derivative at \( x = 0 \):
\[ f''(0) = 2 \]

### Step 6: Analyze the results
- Since \( f'(0) = 0 \) and \( f''(0) = 2 > 0 \), this indicates that \( x = 0 \) is a point of local minimum.

### Conclusion
Thus, the function \( f(x) \) has a local minimum at \( x = 0 \).

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