Given the first curve is y = 4x² + 2x – 8

Now applying first derivative, we get

Now applying the sum rule of differentiation and the differentiation of the constant term is 0 we get

Now applying the power rule of differentiation we get

This is the slope of the first curve; let this be equal to m₁.

⇒ m₁=8x+2……(i)

Given the second curve is y = x³ – x + 13

Now applying first derivative, we get

Now applying the sum rule of differentiation and the differentiation of the constant term is 0 we get

Now applying the power rule of differentiation we get

This is the slope of the second curve; let this be equal to m₂.

⇒ m₂=3x²-1……(ii)

Now when the curve touch each other, there slope must be equal, i.e.,

m₁=m₂

⇒ 8x+2=3x²-1

⇒ 3x²-8x-1-2=0

⇒ 3x²-8x-3=0

Splitting the middle term, we get

⇒ 3x²+x-9x -3=0

⇒ x(3x+1)-3(3x+1)=0

⇒ (3x+1)(x-3)=0

⇒ (3x+1)=0 or (x-3)=0

Substituting in both the curves equation we get

For first curve, y = 4x² + 2x – 8

For second curve, y = x³ – x + 13

Thus at both the curves do not touch

Substituting x=3 in both the curves equation we get

For first curve, y = 4x² + 2x – 8

⇒y=4(3)²+2(3)-8

⇒y=4(9)+6-8=34

For second curve, y = x³ – x + 13

⇒y= (3)³ – (3) + 13

⇒y=27-3+13=37

Thus at x=3 both the curves do not touch

So the curves y = 4x² + 2x – 8 and y = x³ – x + 13 do not touch each other.