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

question 99

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.
-Consider the quartic function $f ( x ) = a x ^ { 4 } + b x ^ { 3 } + c x ^ { 2 } + d x + e , a \neq 0$ . Must this function have at least one critical point? Give reasons for your answer. (Hint: Must $f ^ { \prime } ( x ) = 0$ for some $x$ ?) How many local extreme values can $f$ have?

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Definitions:

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Virchow's Triad

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Hypercoagulability

An increased tendency for the blood to clot, which can elevate the risk of thrombosis or other cardiovascular problems.