f(x) = x4 − 4x3 + 7

Question

Find the coordinates of points where the graph of f(x) = x⁴ − 4x³ + 7 has horizontal tangents. As a check, graph f(x) and see whether the points you found look as though they have horizontal tangents.

(x, y) = ( ) (Smaller x- value)

(x, y) = ( ) (Larger x- value)

Answer 1

f'(x) = 4x³ - 12x² = 0, 4x²(x - 3) = 0, x = 0 and x = 3

f(0) = 7, f(3) = -20

(0,3) and (3, -20) two points where tangent line horizontal.