Problem: Where is the tangent line to the graph of

f (x) = x³ − 3x² − 9x + 11

horizontal?

Solution: A line is horizontal exactly if its slope is zero. The slope of the tangent line at some point a is given by k_T = f ′(a), so the points with horizontal tangent lines are exactly those that satisfy the equation f ′(x) = 0. We solve this equation:

3x² − 6x − 9 = 0, that is, 3(x + 1)⋅(x − 3) = 0.

The points where the tangent line is horizontal are a = −1 and a = 3.

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