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).