How do you find the critical numbers of the function gy = y^2 + 2y + 4?

To find the critical numbers of the function g(y) = y² + 2y + 4, we need to follow a few key steps. Critical numbers occur where the derivative equals zero or where the derivative does not exist.

Step 1: Compute the derivative.

The first step is to calculate the derivative of the function with respect to y. Using the power rule:

g'(y) = 2y + 2

Step 2: Set the derivative equal to zero.

Next, we will set the derivative equal to zero in order to find the points where the slope of the tangent line is horizontal:

2y + 2 = 0

Now, solve for y:

2y = -2
y = -1

Step 3: Check for where the derivative does not exist.

In this case, since g'(y) = 2y + 2 is a polynomial function, it exists for all values of y. Therefore, there are no points where the derivative does not exist.

Conclusion: List the critical numbers.

From our analysis, the only critical number for the function g(y) = y² + 2y + 4 is:

y = -1

This indicates the point where the function may have a local maximum or minimum, or a point of inflection. To determine the nature of this critical number, you can use the second derivative test or analyze the first derivative around that point.