`f(x) = (x - 4)/(x^2 - 7)` Determine the point(s) at which the graph of the function has a horizontal tangent line.

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Chapter 2, 2.3 - Problem 76 - Calculus of a Single Variable (10th Edition, Ron Larson).
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mathace | (Level 3) Assistant Educator

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Given: `f(x)=(x-4)/(x^2-7)`

Find the derivative of the function using the Quotient Rule. Set the derivative equal to zero and solve for the critical x value(s). When the derivative is zero the slope of the tangent line will be horizontal to the graph.

`f'(x)=[(x^2-7)(1)-(x-4)(2x)]/(x^2-7)^2=0`

`(x^2-7-2x^2+8x)=0`

`-x^2+8x-7=0`

`x^2-8x+7=0`

`(x-1)(x-7)=0`

`x=1,x=7`

Plug in the critical values for x into the f(x) equation.

`f(x)=(x-4)/(x^2-7)`

`f(1)=(1-4)/(1^2-7)=-3/-6=1/2`

`f(7)=(7-4)/(7^2-7)=3/42=1/14` 

The tangent line will be horizontal to the graph at points `(1,1/2)`  and `(7,1/14).`

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