Question

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.)

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:

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