Let f(x) = x3 − 2x2. Find the points on the graph of f where the tangent line is horizontal. (Enter your answers as a comma-separated list.)

Question

The larger x value (x,y)

Answer 1

The slope of the tangent line at (x,y) is f'(x)

f'(x) = 3x² - 4x

Tangent line is horizontal when slope = 0

So we need f'(x) = 0 [basically finding critical points ]

so 3x² - 4x = 0, so x (3x-4) = 0, so x= 0 or 4/3

x = 0 f(x) = 0

x = 4/3, f(x) = (64/27) - 2 (16/9) = (64/27) - (96/27) = -32/27

So the points are (0,0) and (4/3, -32/27)