Find the points of local maxima and local minima, if any, of the following functions. Find also the local maximum and local minimum values :
`f(x)=x^(2)`

Answer

To find the points of local maxima and local minima for the function \( f(x) = x^2 \), we will follow these steps:

### Step 1: Find the first derivative of the function
We start by differentiating the function \( f(x) \):
\[ f'(x) = \frac{d}{dx}(x^2) = 2x \]

### Step 2: Set the first derivative equal to zero
To find the critical points, we set the first derivative equal to zero:
\[ 2x = 0 \]
Solving for \( x \), we get:
\[ x = 0 \]

### Step 3: Find the second derivative of the function
Next, we differentiate the first derivative to find the second derivative:
\[ f''(x) = \frac{d}{dx}(2x) = 2 \]

### Step 4: Determine the nature of the critical point
Now, we evaluate the second derivative at the critical point \( x = 0 \):
\[ f''(0) = 2 \]
Since \( f''(0) > 0 \), this indicates that the function has a local minimum at \( x = 0 \).

### Step 5: Find the local minimum value
To find the local minimum value, we substitute \( x = 0 \) back into the original function:
\[ f(0) = 0^2 = 0 \]

### Conclusion
Thus, the function \( f(x) = x^2 \) has a local minimum at the point \( (0, 0) \) with a minimum value of \( 0 \).

### Summary:
- Local minimum point: \( (0, 0) \)
- Local minimum value: \( 0 \)

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