Q. 5 C5.0( 2 Votes )

At what points on the following curves, is the tangent parallel to the x–axis?

y = 12(x + 1) (x – 2) on [–1, 2]

Answer :

First, let us write the conditions for the applicability of Rolle’s theorem:


For a Real valued function ‘f’:


a) The function ‘f’ needs to be continuous in the closed interval [a,b].


b) The function ‘f’ needs differentiable on the open interval (a,b).


c) f(a) = f(b)


Then there exists at least one c in the open interval (a,b) such that f’(c) = 0.


Given function is:


y = 12(x + 1)(x – 2) on [ – 1,2]


We know that polynomials are continuous and differentiable over R.


Let’s check the values of y at the extremums


y( – 1) = 12( – 1 + 1)( – 1 – 2)


y( – 1) = 12(0)( – 3)


y( – 1) = 0


y(2) = 12(2 + 1)(2 – 2)
y(2) = 12(3)(0)


y(2) = 0


We got y( – 1) = y(2). So, there exists a c such that f’(c) = 0.


For a curve g to have a tangent parallel to the x – axis at point r, the criteria to be satisfied is g’(r) = 0.


y’(x) = 0




((x + 1)×1) + ((x – 2)×1) = 0


x + 1 + x – 2 = 0


2x – 1 = 0


2x = 1



The value of y is




y = – 27


The point at which the curve has tangent parallel to x – axis is .


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