Calculus Question

Question

Find the critical numbers of the function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) g(y)=(y-1)/(y2 − 3y + 3)

 

Answer 1

Critical points are values that make the derivative equal to zero, so the first step here is to find the derivative. We'll have to make use of the quotient rule:

g'(y) = ( (y² - 3y + 3) - (y - 1)(2y - 3) ) / ( y² - 3y + 3)²

g'(y) = ( y² - 3y + 3 - 2y² + 5y - 3 ) / ( y² - 3y + 3)²

g'(y) = ( -y² + 2y ) / ( y² - 3y + 3)²

g'(y) = -y ( y - 2 )/ ( y² - 3y + 3)²

So if we set g'(y) = 0 and solve for y, we find that y can be either 0 or 2. So the answer is 0, 2.