Find Intervals of Increase for y = x² + 4x + 5

To solve the problem and find the intervals of increase for the function $y = x^2 + 4x + 5$, we need to follow these steps:

• Step 1: Calculate the derivative of the function.
• Step 2: Determine where the derivative is greater than zero.
• Step 3: Identify the interval where the function is increasing.

Let's perform each step in detail:

Step 1: Calculate the derivative of the function $y = x^2 + 4x + 5$.
The derivative of $y$ with respect to $x$, denoted as $y'$, is obtained by differentiating each term:
$y' = \frac{d}{dx}(x^2) + \frac{d}{dx}(4x) + \frac{d}{dx}(5)$.
This gives $y' = 2x + 4$.

Step 2: Determine where the derivative is greater than zero.
We solve the inequality $2x + 4 > 0$.
Subtract 4 from both sides: $2x > -4$.
Divide by 2: $x > -2$.

Step 3: Identify the interval where the function is increasing.
The function increases on the interval $x > -2$.

Therefore, the solution to the problem is that the function is increasing for $x > -2$.