Answer :

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.


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