At which point(s) does the graph of the function #f(x) = (x^2) / (x - 1)# have a horizontal tangent line?

Answer 1

#(0, 0), (2, 4)#

The slope of the tangent line of a graph #y=f(x)# at a point #x_0# is given by the derivative of #f# at that point, that is, #f'(x_0)#.
A horizontal tangent line implies a slope of #0#, so our goal is to find the points at which the derivative #f(x)# evaluates to #0#.

Using the quotient rule, we find the derivative as

#f'(x) = d/dx x^2/(x-1)#
#=((x-1)(d/dxx^2)-x^2(d/dx(x-1)))/(x-1)^2#
#=(2x(x-1)-x^2(1))/(x-1)^2#
#=(2x^2-2x-x^2)/(x-1)^2#
#=(x^2-2x)/(x-1)^2#
#=(x(x-2))/(x-1)^2#

Setting this equal to zero, we get

#(x(x-2))/(x-1)^2 = 0#
#=> x(x-2) = 0#
#=> x = 0 or x = 2#
Thus, the graph of #f(x)# has a horizontal tangent line at #x=0# and #x=2#, that is, at the points
#(0, 0), (2, 4)#
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Answer 2

The graph of the function f(x) = (x^2) / (x - 1) has a horizontal tangent line at x = 2.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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