Find the critical numbers of the function. g(y) = (y - 2) / (y² − 2y + 4).

Solution:

A critical point of a function of f(x), is a value x0 in the domain of f where it is not differentiable or its derivative is zero

i.e. (f ′(x0) = 0).

If f has a local maximum or minimum at c, then c is a critical number of the function f.

Given, g(y) = (y - 2) / (y² − 2y + 4)

Taking derivative,

g'(y) = (y² − 2y + 4)(1) - (y - 2) (2y - 2) / (y² − 2y +4 )²

g'(y) = (y² − 2y + 4) - (2y² - 2y - 4y + 4) / (y² - 2y + 4)²

On simplification,

g'(y) = (y² - 2y + 4 - 2y² + 6y - 4) / (y² - 2y + 4)²

g'(y) = (-y² + 4y ) / (y² - 2y + 4)²

Taking out common term,

g'(y) = y(4 - y) / (y² - 2y + 4)²

To find critical number, g'(y) = 0

y(4 - y) / (y² - 2y + 4)² = 0

y(4 - y) = 0

y = 0

(4 - y) = 0

y = 4

Therefore, the critical numbers are y = 0 and y = 4.

---

Find the critical numbers of the function. g(y) = (y - 2) / (y² − 2y + 4).

Summary:

The critical numbers of the function g(y) = (y - 2) / (y² − 2y + 4) are y = 0 and y = 4.