Find the point(s) on the graph of f where the tangent line is horizontal. (If an answer does not exist, enter DNE.) f(x) = x/x^2+9

Question

(x,y) (_________) (smaller x-value)

(x,y) (_________) (larger x-value)

Answer 1

Find the derivative of the function and set f ' (x) equal to 0 since the tangent line is horizontal, at that ----slope will be zero.

f (x) = x / (x²+9)

you will find 2 or 1 x value depending on the equations. take those X values and plugin into original equation f(x) to find y value to find the ordered pair (X,Y)

Find f ' (x) using the quotient rule

(x²+9) (1) - (x)(2x) / (x²+9)²

(x²+9) - (2x²) / (x²+9)²

(x²+9) - (2x²) / (x²+9)² = 0 drop the (x²+9)² since you will mulitply over it will gone

(x²+9) - (2x²) = 0

x²+9 - 2x² = 0

-x²+9 = 0

-(x² - 9 ) = 0

-(x-3)(x+3) = )

x = 3 and x = -3

plug into the original function to find the values

you will have 2 answer

( -3, -1/6) small one

( 3, 1/6 ) large one