find the value or values of c that satisfy the equation \\( \\frac { f ( b ) - f ( a ) } { b - a } = f ^ {…

find the value or values of c that satisfy the equation \\( \\frac { f ( b ) - f ( a ) } { b - a } = f ^ { prime } ( c ) \\) in the conclusion of the mean value theorem for the given function on the given interval. \n\n\\( f ( x ) = x ^ { \frac { 7 } { 4 } }, 0,1 \\)\n\n\\( c = \\) (use a comma to separate answers as needed.)

find the value or values of c that satisfy the equation \\( \\frac { f ( b ) - f ( a ) } { b - a } = f ^ { prime } ( c ) \\) in the conclusion of the mean value theorem for the given function on the given interval. \n\n\\( f ( x ) = x ^ { \frac { 7 } { 4 } }, 0,1 \\)\n\n\\( c = \\) (use a comma to separate answers as needed.)

Answer

Explanation:

Step1: Calculate (f(a)) and (f(b))

Given (a = 0), (b = 1), (f(x)=x^{\frac{7}{4}}). (f(a)=f(0)=0^{\frac{7}{4}} = 0) (f(b)=f(1)=1^{\frac{7}{4}}=1)

Step2: Calculate (\frac{f(b)-f(a)}{b - a})

(\frac{f(1)-f(0)}{1 - 0}=\frac{1 - 0}{1}=1)

Step3: Find (f^{\prime}(x))

Using the power rule ((x^{n})^\prime=nx^{n - 1}), for (f(x)=x^{\frac{7}{4}}), (f^{\prime}(x)=\frac{7}{4}x^{\frac{7}{4}-1}=\frac{7}{4}x^{\frac{3}{4}})

Step4: Solve (f^{\prime}(c)=\frac{f(b)-f(a)}{b - a})

Set (\frac{7}{4}c^{\frac{3}{4}}=1) (c^{\frac{3}{4}}=\frac{4}{7}) Raise both sides to the (\frac{4}{3}) power: (c = (\frac{4}{7})^{\frac{4}{3}}=\frac{4^{\frac{4}{3}}}{7^{\frac{4}{3}}}=\frac{2^{2\times\frac{4}{3}}}{7^{\frac{4}{3}}}=\frac{2^{\frac{8}{3}}}{7^{\frac{4}{3}}}=\frac{16^{\frac{1}{3}}}{49^{\frac{1}{3}}}=\sqrt[3]{\frac{16}{49}})

Answer:

(\sqrt[3]{\frac{16}{49}})