How do you determine if Rolle's Theorem can be applied to the given functions #f(x) = x^4 -2x^2# on interval [-2,2]?

Question

Christopher Allers · Accepted Answer

See the Explanation section.

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 of Rolle's Theorem are true for the function #f(x) = x^4-2x^2# on the interval #[-2,2]#. 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 #[-2,2]# [Because, this function is a polynomial so it is continuous at every real number.] H2 : The function #f# in this problem is differentiable on #(-2,2)# [Because the derivative, #f'(x) = 4x^3-4x# exists for all real #x#. In particular, it exists for all #x# in #(-2,2)#.) H3 : #f(-2) = f(2)# either by observing that #f# is an even function or by arithmetic (#f(-2) = 8# and #f(2) = 8# Therefore we can apply Rolle's Theorem to #f(x) = x^4-2x^2# on the interval #[-2,2]#. (Meaning "the hypotheses are true".)

Christopher Agresta · Answer

undefined To determine if Rolle's Theorem can be applied to the given function $ f(x) = x^4 - 2x^2 $ on the interval $[-2, 2]$, you need to check two conditions:

1. The function must be continuous on the closed interval $[-2, 2]$.
2. The function must be differentiable on the open interval $(-2, 2)$.

Since $ f(x) = x^4 - 2x^2 $ is a polynomial function, it is continuous and differentiable everywhere. Therefore, Rolle's Theorem can be applied to the given function on the interval $[-2, 2]$.