The minimum value of `(x)^6`

Explanation

To find the minimum value of \( f(x) = x^6 \), we can follow these steps:

### Step 1: Define the function
Let \( f(x) = x^6 \).

### Step 2: Differentiate the function
To find the critical points, we need to differentiate \( f(x) \):
\[ f'(x) = \frac{d}{dx}(x^6) = 6x^5 \]

### Step 3: Set the derivative to zero
To find the critical points, set the first derivative equal to zero:
\[ 6x^5 = 0 \]
This simplifies to:
\[ x^5 = 0 \]
Thus, we find:
\[ x = 0 \]

### Step 4: Determine the nature of the critical point
To determine if this critical point is a minimum or maximum, we can use the second derivative test. First, we differentiate \( f'(x) \):
\[ f''(x) = \frac{d}{dx}(6x^5) = 30x^4 \]

Now, evaluate the second derivative at the critical point \( x = 0 \):
\[ f''(0) = 30(0^4) = 0 \]
Since the second derivative is zero, the test is inconclusive. We need to analyze the first derivative around the critical point.

### Step 5: Analyze the first derivative
We will check the sign of \( f'(x) \) around \( x = 0 \):
- For \( x < 0 \) (e.g., \( x = -1 \)):
\[ f'(-1) = 6(-1)^5 = -6 \quad (\text{negative}) \]
- For \( x > 0 \) (e.g., \( x = 1 \)):
\[ f'(1) = 6(1)^5 = 6 \quad (\text{positive}) \]

Since \( f'(x) \) changes from negative to positive at \( x = 0 \), this indicates that \( x = 0 \) is a point of local minimum.

### Step 6: Find the minimum value
Now, we can find the minimum value of the function at this point:
\[ f(0) = 0^6 = 0 \]

### Conclusion
The minimum value of \( x^6 \) is \( 0 \) at \( x = 0 \).

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