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 { 7 } { 4 } } ; 0,1 \\) \\( c = \\sqrt 3 { \\frac { 16 } { 49 } } \\) (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 { 7 } { 4 } } ; 0,1 \\) \\( c = \\sqrt 3 { \\frac { 16 } { 49 } } \\) (use a comma to separate answers as needed.)

Answer

Explanation:

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

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

Step2: Find the derivative ( 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}} ).

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

Set ( \frac{7}{4}c^{\frac{3}{4}}=1 ). First, multiply both sides by ( \frac{4}{7} ): ( c^{\frac{3}{4}}=\frac{4}{7} ). Then, raise both sides to the ( \frac{4}{3} ) power: ( c=\left(\frac{4}{7}\right)^{\frac{4}{3}}=\sqrt[3]{\left(\frac{4}{7}\right)^4}=\sqrt[3]{\frac{256}{2401}}=\sqrt[3]{\frac{16\times16}{49\times49}}=\sqrt[3]{\frac{16}{49}\times\frac{16}{49}}=\sqrt[3]{\frac{16}{49}} ).

Answer:

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