Find the Value or Values Of $c$ That Satisfy the Equation $\frac { f ( b ) - f ( a ) } { b - a } = f ^ { \prime } ( c )$

question 84

Essay

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 function and interval.
-Let $f ( x ) = ( x - 7 ) ^ { 2 / 3 }$
(a) Does $f ^ { \prime } ( 7 )$ exist?
(b) Show that the only local extreme value of $f$ occurs at $x = 7$ .
(c) Does the result of (b) contradict the Extreme Value Theorem?
(d) Repeat parts (a) and (b) for $\mathrm { f } ( \mathrm { x } ) = ( \mathrm { x } - \mathrm { c } ) ^ { 2 / 3 }$ .
Give reasons for your answers.