Tangent line is horizontal

Question

Let f(x) = x3 − 4x2.

 

Find the point(s) on the graph of f where the tangent line is horizontal.
(x, y) =____________ (smaller x-value)
(x, y) =____________ (larger x-value)

Answer 1

f(x) = x³ - 4x²

 

f'(x) = 3x² - 8x = x(3x-8)

 

Horizontal tangent when f'(x) = 0

 

                                       x = 0, 8/3

 

Horizontal tangent at (0, f(0)) and (8/3, f(8/3))

 

                            = (0,0) and (8/3, -256/27)

 

Answer 2

f(x)=x³-4x²

If x₀ is tangent point, then f'(x₀)=k, where k is the slope (y=kx+b). The tangent is hozirontal if k=0, or this is the same if f'(x₀)=0. Therefore we have to find such x: f'(x)=0

f'(x)=3x²-8x=0

x=0, x=8/3.

if x=0, then y=0.

if x=8/3, then y=(8/3)^3--4(8/3)²=-256/27

(x, y) =(0, 0) smaller

(x, y)=(8/3, -256/27) larger