Determine the intervals in which the following functions are strictly increasing or strictly decreasing :
`f(x)=x^(2)+2x-5`

Answer

To determine the intervals in which the function \( f(x) = x^2 + 2x - 5 \) 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, \( f'(x) \).

\[ f(x) = x^2 + 2x - 5 \]

Differentiating \( f(x) \):

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

### Step 2: Set the derivative equal to zero to find critical points
Next, we need to find the critical points by setting the derivative equal to zero:

\[ 2x + 2 = 0 \]

Solving for \( x \):

\[ 2x = -2 \]
\[ x = -1 \]

### Step 3: Determine the sign of the derivative in the intervals
Now we will analyze the sign of \( f'(x) \) in the intervals determined by the critical point \( x = -1 \). We will test values in the intervals \( (-\infty, -1) \) and \( (-1, \infty) \).

1. **Interval \( (-\infty, -1) \)**: Choose \( x = -2 \)

\[ f'(-2) = 2(-2) + 2 = -4 + 2 = -2 \quad (\text{negative}) \]

2. **Interval \( (-1, \infty) \)**: Choose \( x = 0 \)

\[ f'(0) = 2(0) + 2 = 0 + 2 = 2 \quad (\text{positive}) \]

### Step 4: Determine the intervals of increase and decrease
From our analysis:

- \( f'(x) < 0 \) in the interval \( (-\infty, -1) \) means the function is **strictly decreasing** on this interval.
- \( f'(x) > 0 \) in the interval \( (-1, \infty) \) means the function is **strictly increasing** on this interval.

### Conclusion
Thus, the function \( f(x) = x^2 + 2x - 5 \) is:
- **Strictly decreasing** on the interval \( (-\infty, -1) \)
- **Strictly increasing** on the interval \( (-1, \infty) \)