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

on [–1, 1]

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:


on [ – 1,1]


We know that exponential functions are continuous and differentiable over R.


Let’s check the values of y at the extremums



y( – 1) = e1 – 1


y( – 1) = e0
y( – 1) = 1



y(1) = e1 – 1


y(1) = e0


y(1) = 1


We got y( – 1) = y(1). 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





2x = 0


x = 0


The value of y is



y = e1 – 0


y = e1


y = e


The point at which the curve has a tangent parallel to the x – axis is (0,e).


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