For what values of x, the function `f(x) = x^(5)-5x^(4)+5x(3)-1` is maximum or minimum? Prove that at x = 0, the function is neither maximum nor minimum.

Answer

To find the values of \( x \) for which the function \( f(x) = x^5 - 5x^4 + 5x^3 - 1 \) has maximum or minimum points, we need to follow these steps:

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

\[ f'(x) = \frac{d}{dx}(x^5 - 5x^4 + 5x^3 - 1) \]

Using the power rule, we get:

\[ f'(x) = 5x^4 - 20x^3 + 15x^2 \]

### Step 2: Set the first derivative to zero
To find the critical points, we set the first derivative equal to zero:

\[ 5x^4 - 20x^3 + 15x^2 = 0 \]

Factoring out the common term \( 5x^2 \):

\[ 5x^2(x^2 - 4x + 3) = 0 \]

### Step 3: Solve for \( x \)
This gives us two factors:

1. \( 5x^2 = 0 \) which gives \( x = 0 \)
2. \( x^2 - 4x + 3 = 0 \)

Now, we can factor the quadratic:

\[ (x - 1)(x - 3) = 0 \]

Thus, we have the critical points:

\[ x = 0, \quad x = 1, \quad x = 3 \]

### Step 4: Find the second derivative
Next, we find the second derivative to determine the nature of these critical points:

\[ f''(x) = \frac{d}{dx}(5x^4 - 20x^3 + 15x^2) \]

Calculating the second derivative:

\[ f''(x) = 20x^3 - 60x^2 + 30x \]

### Step 5: Evaluate the second derivative at critical points
We will evaluate \( f''(x) \) at each critical point:

1. **At \( x = 0 \)**:
\[ f''(0) = 20(0)^3 - 60(0)^2 + 30(0) = 0 \]

2. **At \( x = 1 \)**:
\[ f''(1) = 20(1)^3 - 60(1)^2 + 30(1) = 20 - 60 + 30 = -10 \quad (\text{Negative, so maximum}) \]

3. **At \( x = 3 \)**:
\[ f''(3) = 20(3)^3 - 60(3)^2 + 30(3) = 540 - 540 + 90 = 90 \quad (\text{Positive, so minimum}) \]

### Conclusion
- The function has a maximum at \( x = 1 \) (since \( f''(1) < 0 \)).
- The function has a minimum at \( x = 3 \) (since \( f''(3) > 0 \)).
- At \( x = 0 \), since \( f''(0) = 0 \), we cannot conclude if it is a maximum or minimum. We can also check the values of \( f(x) \) around \( x = 0 \) to confirm that it is neither a maximum nor a minimum.

### Summary
The values of \( x \) for which the function has maximum or minimum are:
- Maximum at \( x = 1 \)
- Minimum at \( x = 3 \)
- At \( x = 0 \), the function is neither maximum nor minimum.