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

question 51

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 $f ( x ) = \left| x ^ { 3 } - 9 x \right|$
(a) Does $f ^ { \prime } ( 0 )$ exist?
(b) Does $f ^ { \prime } ( 3 )$ exist?
(c) Does $f ^ { \prime } ( - 3 )$ exist?
(d) Determine all extrema of $f$ .

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Definitions:

Common Hepatic Duct

A duct formed by the convergence of the right and left hepatic ducts that transport bile from the liver to the gallbladder and duodenum.

Cystic Duct

A small duct that connects the gallbladder to the common bile duct, allowing bile to flow into and out of the gallbladder.

Pancreatic Duct

A duct connecting the pancreas to the common bile duct to supply pancreatic juices containing digestive enzymes to the small intestine.

Parotid Duct

A tube that transports saliva from the parotid gland into the mouth, helping with digestion.