Question

Examine the validity of the hypothesis and the conclusion of Rolle's Theorem: f(x)= x ^3- 4x on [-2, 2].Please explain it.​

Answer 1

Answer:

he function f(x)=x

3

−4x is a polynomial and so it is continuous and differentiable at all xϵ R.

In particular it is continuous in the closed interval [−2,2] .

Also f(−2)]=0=f(2). Thus , f(x) satisfies all three conditions of Rolle's theorem in (−2,2).

Therefore , there must exist at least one real number

′

x

′

in the

open interval (−2,2) for which f

′

(x)=0

Also f

′

(x)=3x

2

−4

Now f

′

(x)=0 gives 3x

2

−4=0 or x=±

3

2

which is also known as stationary point.

Both these value lie in the open interval (−2,2) and thus the conclusion of Rolle's theorem is verified