Determine the intervals in which the following functions are strictly increasing or strictly decreasing :
`f(x)=x^(8)+6x^(2)`.

Answer

To determine the intervals in which the function \( f(x) = x^8 + 6x^2 \) is strictly increasing or strictly decreasing, we will follow these steps:

### Step 1: Find the derivative of the function
To analyze the behavior of the function, we first need to find its derivative:
\[ f'(x) = \frac{d}{dx}(x^8 + 6x^2) \]
Using the power rule, we differentiate each term:
\[ f'(x) = 8x^7 + 12x \]

### Step 2: Set the derivative equal to zero
Next, we find the critical points by setting the derivative equal to zero:
\[ 8x^7 + 12x = 0 \]
We can factor out the common term:
\[ 4x(2x^6 + 3) = 0 \]
This gives us two factors:
1. \( 4x = 0 \) which leads to \( x = 0 \)
2. \( 2x^6 + 3 = 0 \) which has no real solutions since \( 2x^6 \) is always non-negative and cannot equal -3.

Thus, the only critical point is \( x = 0 \).

### Step 3: Determine the sign of the derivative
To determine where the function is increasing or decreasing, we need to analyze the sign of \( f'(x) \) around the critical point \( x = 0 \).

- For \( x < 0 \) (e.g., \( x = -1 \)):
\[ f'(-1) = 8(-1)^7 + 12(-1) = -8 - 12 = -20 < 0 \]
This means \( f(x) \) is decreasing on the interval \( (-\infty, 0) \).

- For \( x > 0 \) (e.g., \( x = 1 \)):
\[ f'(1) = 8(1)^7 + 12(1) = 8 + 12 = 20 > 0 \]
This means \( f(x) \) is increasing on the interval \( (0, \infty) \).

### Step 4: Conclusion
Based on our analysis, we can conclude:
- The function \( f(x) \) is **strictly decreasing** on the interval \( (-\infty, 0) \).
- The function \( f(x) \) is **strictly increasing** on the interval \( (0, \infty) \).

### Summary of Intervals
- Strictly Decreasing: \( (-\infty, 0) \)
- Strictly Increasing: \( (0, \infty) \)