If the function `f(x) = x^(2) + 2x -5` is increasing function then `x lt -1`.

Answer

To determine whether the function \( f(x) = x^2 + 2x - 5 \) is an increasing function when \( x < -1 \), we need to analyze the derivative of the function. Here’s the step-by-step solution:

### Step 1: Find the derivative of the function
The first step is to differentiate the function \( f(x) \).

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

### Step 2: Calculate the derivative
Using the power rule of differentiation, we find:

\[ f'(x) = 2x + 2 \]

### Step 3: Set the derivative greater than zero
For the function to be increasing, we need the derivative to be greater than zero:

\[ f'(x) > 0 \]

Substituting the expression we found for \( f'(x) \):

\[ 2x + 2 > 0 \]

### Step 4: Solve the inequality
Now, we solve the inequality for \( x \):

\[ 2x > -2 \]
\[ x > -1 \]

### Step 5: Conclusion
From the inequality \( x > -1 \), we conclude that the function \( f(x) \) is increasing for all \( x \) greater than \(-1\). Therefore, the statement that the function is increasing when \( x < -1 \) is **false**.

### Summary
The function \( f(x) = x^2 + 2x - 5 \) is increasing for \( x > -1 \), not for \( x < -1 \).

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