Determine the points at which the graph of the function has a horizontal tangent line. f(x) = x² / x - 8?

Solution:

Given, f(x) = x² / x - 8

We have to find the points at which the graph of the function has a horizontal tangent line.

A point on a function will have a horizontal tangent line where the first derivative is zero.

Thus, f’(x) = 2x/(x - 8) - x² / (x - 8)²

On simplification,

f’(x) = (2x(x - 8) - x²)/(x - 8)²

= 2x² - 16x - x² / (x - 8)²

= x² - 16x / (x - 8)²

Now, f’(x) = 0

x² - 16x / (x - 8)² = 0

x² - 16x = 0

x(x - 16) = 0

So, x = 0

x - 16 = 0

x = 16

Put x = 0 in f(x)

f(16) = (0)² / (0) - 8 = 0

Put x = 16 in f(x)

f(16) = (16)² / (16 - 8)

= 256/8

= 32

Therefore, the points are (0, 0) and (16, 32).

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Determine the points at which the graph of the function has a horizontal tangent line. f(x) = x² / x - 8?

Summary:

The points at which the graph of the function has a horizontal tangent line. f(x) = x² / x - 8 are (0, 0) and (16, 32).