This statement is true or false?Please give reasons for your answer. Rolle's Theorem is applicable for the function f,defined by f(x)=1+x^(2/3) in the interval [-1,1] ?

1 Answer
Feb 14, 2018

Please see below.

Explanation:

When we are asked whether some theorem "can be applied" to some situation, we are really being asked "Are the hypotheses of the theorem true for this situation?"

(The hypotheses are also called the antecedent, of 'the if parts'.)

So we need to determine whether the hypotheses ot Rolle's Theorem are true for the function

#f(x) = 1+x^(2/3)# on the interval #[-1,1]#

Rolle's Theorem has three hypotheses:

H1 : #f# is continuous on the closed interval #[a,b]#
H2 : #f# is differentiable on the open interval #(a,b)#.
H3 : #f(a)=f(b)#

We say that we can apply Rolle's Theorem if all 3 hypotheses are true.

H1 : The function #f# in this problem is continuous on #[-1,1]#

H2 : The function #f# in this problem is not differentiable on #(-1,1)# because it is not differentiable at #0#. (The derivative function #2/(3root(3)(x))# is not defined for #x = 0#.)

H3 : #f(-1) = 2 = f(1)#

If even one hypothesis fails to be true, then we cannot apply the theorem.
Therefore we CANNOT apply Rolle's Theorem to #f(x) = 1+x^(2/3)# on the interval #[-1,1]#.

By the way, the conclusion of Rolle's Theorem is also false for this function on this interval. There is no solution to #f'(x) = 2/(3root(3)(x)) = 0#