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. \\( f ( x ) = x ^ { \\frac { 9 } { 7 } } ; 0,1 \\) \\( 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. \\( f ( x ) = x ^ { \\frac { 9 } { 7 } } ; 0,1 \\) \\( c = \\) (use a comma to separate answers as needed.)

Answer

Explanation:

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

Given (f(x)=x^{\frac{9}{7}}), (a = 0), (b = 1). (f(b)-f(a)=1^{\frac{9}{7}}-0^{\frac{9}{7}}=1) (b - a=1 - 0=1) So (\frac{f(b)-f(a)}{b - a}=\frac{1}{1}=1)

Step2: Find (f^{\prime}(x))

Using the power rule ((x^n)^\prime=nx^{n - 1}), for (f(x)=x^{\frac{9}{7}}), (f^{\prime}(x)=\frac{9}{7}x^{\frac{9}{7}-1}=\frac{9}{7}x^{\frac{2}{7}})

Step3: Solve (f^{\prime}(c)=1)

Set (\frac{9}{7}c^{\frac{2}{7}}=1) (c^{\frac{2}{7}}=\frac{7}{9}) Raise both sides to the (\frac{7}{2}) power: (c = (\frac{7}{9})^{\frac{7}{2}}=\frac{7^{\frac{7}{2}}}{9^{\frac{7}{2}}}=\frac{7^{3}\sqrt{7}}{9^{3}\sqrt{9}}=\frac{343\sqrt{7}}{2187})

Answer:

(\frac{343\sqrt{7}}{2187})