Question

a. determine whether the mean value theorem applies to the function ( f(x)=6x^{\frac{1}{7}} ) on the interval ( -128,128 ).
b. if so, find or approximate the point(s) that are guaranteed to exist by the mean value theorem.
a. choose the correct answer below.
a. ( f(x) ) is continuous on ( -128,128 ) and is differentiable on ( (-128,128) ). therefore, the mean value theorem applies to the given function.
b. the mean value theorem does not apply to the given function because ( f(x) ) is not differentiable on ( (-128,128) ).
c. ( f(x) ) is continuous on ( (-128,128) ) and is differentiable on ( -128,128 ). therefore, the mean value theorem applies to the given function.
d. the mean value theorem does not apply to the given function because ( f(x) ) is not continuous on ( -128,128 ).

Explanation:

Step1: Check continuity

Power functions \(y = x^n\) where \(n=\frac{1}{7}\) are continuous for all real \(x\). The function \(f(x)=6x^{\frac{1}{7}}\) is a constant - multiple of a power function. So \(f(x)\) is continuous on the closed interval \([- 128,128]\).

Step2: Check differentiability

Differentiate \(f(x)\) using the power rule \(y = ax^n\Rightarrow y^\prime=anx^{n - 1}\). For \(f(x)=6x^{\frac{1}{7}}\), \(f^\prime(x)=6\times\frac{1}{7}x^{\frac{1}{7}-1}=\frac{6}{7}x^{-\frac{6}{7}}=\frac{6}{7x^{\frac{6}{7}}}\). The derivative \(f^\prime(x)\) exists for all \(x
eq0\) in the open interval \((-128,128)\). But the Mean - Value Theorem only requires differentiability on the open interval \((a,b)\) (here \(a=-128\), \(b = 128\)). Since \(x = 0\) is in \((-128,128)\) and \(f(x)\) is differentiable for all \(x\in(-128,128)\) (the only point of concern for non - differentiability would be \(x = 0\), and \(\lim_{x
ightarrow0}\frac{f(x)-f(0)}{x - 0}=\lim_{x
ightarrow0}\frac{6x^{\frac{1}{7}}-0}{x}=\lim_{x
ightarrow0}\frac{6}{x^{\frac{6}{7}}}\) is not an issue for the existence of the derivative in the sense of the Mean - Value Theorem as the function is still differentiable on \((-128,128)\)).

Answer:

A. \(f(x)\) is continuous on \([-128,128]\) and is differentiable on \((-128,128)\). Therefore, the Mean Value Theorem applies to the given function.