Find the points of local maxima or local minima and the corresponding local maximum and minimum values of each of the following functions:
`f(x) = (x -3)^(4)`

Explanation

To find the points of local maxima or minima for the function \( f(x) = (x - 3)^4 \), we will follow these steps:

### Step 1: Find the first derivative
We start by differentiating the function \( f(x) \).

\[ f'(x) = \frac{d}{dx}[(x - 3)^4] \]

Using the power rule of differentiation, we have:

\[ f'(x) = 4(x - 3)^3 \]

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

\[ 4(x - 3)^3 = 0 \]

Dividing both sides by 4 gives:

\[ (x - 3)^3 = 0 \]

Taking the cube root of both sides, we find:

\[ x - 3 = 0 \implies x = 3 \]

### Step 3: Find the second derivative
Now, we need to find the second derivative to determine whether the critical point is a local maximum or minimum.

\[ f''(x) = \frac{d}{dx}[4(x - 3)^3] \]

Using the power rule again, we get:

\[ f''(x) = 12(x - 3)^2 \]

### Step 4: Evaluate the second derivative at the critical point
We evaluate the second derivative at the critical point \( x = 3 \):

\[ f''(3) = 12(3 - 3)^2 = 12(0)^2 = 0 \]

Since the second derivative is zero, we cannot conclude directly whether it is a maximum or minimum using the second derivative test.

### Step 5: Analyze the first derivative
To determine the nature of the critical point, we can analyze the sign of the first derivative around \( x = 3 \).

- For \( x < 3 \) (e.g., \( x = 2 \)):
\[ f'(2) = 4(2 - 3)^3 = 4(-1)^3 = -4 \quad (\text{negative}) \]

- For \( x > 3 \) (e.g., \( x = 4 \)):
\[ f'(4) = 4(4 - 3)^3 = 4(1)^3 = 4 \quad (\text{positive}) \]

Since \( f'(x) \) changes from negative to positive at \( x = 3 \), this indicates that \( x = 3 \) is a local minimum.

### Step 6: Find the local minimum value
Finally, we find the value of the function at the local minimum:

\[ f(3) = (3 - 3)^4 = 0^4 = 0 \]

### Conclusion
Thus, the function \( f(x) = (x - 3)^4 \) has a local minimum at \( x = 3 \) with a minimum value of \( 0 \).

### Summary
- Local minimum point: \( x = 3 \)
- Local minimum value: \( 0 \)