# Find all points on the curve y=x^3-x+1 where the tangent line is parallel to the line y=2x+5

hala718 | High School Teacher | (Level 1) Educator Emeritus

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Given the curve :

`y= x^3 -x + 1`

`` We need to find the points on the graph such that the tangent lines parallel to the line y= 2x+5

First, we notice that the slope of the line y= 2x+ 5 is 2.

Then, the tangent lines should have  slope of 2.

To find the slope of a tangent line, we need to find the derivative at the point of tendency.

Let us differentitae f(x).

`f'(x)= 3x^2 -1`

`` Let a be the point of tendency such that the tangent line at the point x=a, is parallel to the line 2x+5.

Then the slope is f'(a)= 2

`==> f'(a)= 3a^2 -1 = 2`

`==> 3a^2 = 3`

`==> a^2 = 1`

`a= +-1 `

Now we will find the values of f(x) at the tangent points.

`==> f(1)= 1^3 - 1 +1 = 1`

`f(-1)= -1^3 +1 +1 = 1```

Then, we have two points such that the tangent lines are parallel to the line y= 2x+5 .

Then, the points are: (1, 1) and (-1, 1).