How do you find the points where the graph of the function #y = x - (x / 18)^2# has horizontal tangents and what is the equation?

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Answer 1 · 2016-02-11T16:05:02

#y=81# at #(162,81)#

First, we can simplify the function.

#y=x-x^2/324#

We can find the derivative of this through the power rule:

#y'=1-x/162#

Horizontal tangents will occur when #y'=0#.

#1-x/162=0#

#x/162=1#

#x=162#

We can find the equation of the tangent line by plugging in #162# to the original equation.

#y=162-(162/18)^2#

#y=162-81#

#y=81#

Graphed are the tangent line and original function:

graph{(x-(x/18)^2-y)(y-81+0x)=0 [-119.8, 488.8, -146.2, 158]}