a. determine whether the mean value theorem applies to the function f(x)=6x^(1/9) on the interval -512,512…

a. determine whether the mean value theorem applies to the function f(x)=6x^(1/9) on the interval -512,512. 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 (-512,512) and is differentiable on -512,512. 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 (-512,512). c. the mean value theorem does not apply to the given function because f(x) is not continuous on -512,512. d. f(x) is continuous on -512,512 and is differentiable on (-512,512). therefore, the mean value theorem applies to the given function.

a. determine whether the mean value theorem applies to the function f(x)=6x^(1/9) on the interval -512,512. 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 (-512,512) and is differentiable on -512,512. 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 (-512,512). c. the mean value theorem does not apply to the given function because f(x) is not continuous on -512,512. d. f(x) is continuous on -512,512 and is differentiable on (-512,512). therefore, the mean value theorem applies to the given function.

Answer

Explanation:

Step1: Recall Mean - Value Theorem conditions

The Mean - Value Theorem states that if (y = f(x)) is continuous on the closed interval ([a,b]) and differentiable on the open interval ((a,b)), then there exists at least one (c\in(a,b)) such that (f^{\prime}(c)=\frac{f(b)-f(a)}{b - a}). For the function (f(x)=6x^{\frac{1}{9}}), the domain of (y = x^{\frac{1}{9}}=\sqrt[9]{x}) is all real numbers, so (f(x)) is continuous on ([-512,512]). The derivative of (f(x)) using the power rule ((x^n)^\prime=nx^{n - 1}) is (f^{\prime}(x)=6\times\frac{1}{9}x^{\frac{1}{9}-1}=\frac{2}{3}x^{-\frac{8}{9}}=\frac{2}{3x^{\frac{8}{9}}}). The function (f^{\prime}(x)) is not defined at (x = 0), but (0\in(-512,512)). So (f(x)) is not differentiable on ((-512,512)).

Step2: Determine the correct option

Since (f(x)) is continuous on ([-512,512]) but not differentiable on ((-512,512)), the Mean - Value Theorem does not apply to the given function.

Answer:

C. The Mean Value Theorem does not apply to the given function because (f(x)) is not differentiable on ((-512,512)).