Find all values of x where the tangent lines to y=x^7 and y=x^8 are parallel.

Answer 1

The tangent lines will be parallel when slopes are equal.

The derivatives give formulas for the slopes of tangent lines, so we want to know when are the derivatives equal to each other.

8x⁷ = 7x⁶

8x⁷- 7x⁶=0

x⁶(8x-7) = 0

x = 0 or x = 7/8

The following graph depicts the tangent lines at x = 7/8.

desmos.com/calculator/tl8bvatnr9

Answer 2

Given two lines: y = x^7 and y = x^8

Find where their tangent lines are parallel.

Solution:

If their tangent lines are parallel,

then their derivatives are equal.

So, set the derivatives of each line equal to each other.

d/dx [ x^7 ] = d/dx [ x^8 ]

7*x^6 = 8*x^7

7*x^6 - 8*x^7 = 0

x^6 * ( 7 - 8x ) = 0

x = 0, 7/8

The two lines have parallel tangent lines

at x = 0 and at x = 7/8