determine the values of x, if any, at which f(x)=x^6-x^4 has a horizontal tangent line.

Question

 determine the values of x, if any, at which f(x)=x^6-x^4 has a horizontal tangent line.

Answer 1

Horizontal tangent lines occur where the first derivative of f(x) equals zero

 

f(x) = x⁶ - x⁴

df(x)/dx= 6x⁵ - 4x³ = 2x³(3x² - 2)

0 = 2x³(3x² - 2)

 

This is true when x = 0 or when (3x² - 2) = 0

 

(3x² - 2)= 0

3x² = 2

x² = 2/3

x = ±√(2/3)

 

There are three values of x where f(x) has horizontal tangents: x = 0 and x = ±√(2/3)