Answer the Problem $f ( x ) = a x ^ { 4 } + b x ^ { 3 } + c x ^ { 2 } + d x + e , a \neq 0$

question 118

Essay

Answer the problem.
-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:

Vastus Intermedius

A deep muscle in the anterior compartment of the thigh, part of the quadriceps femoris group, lying between the vastus lateralis and vastus medialis.

Quadriceps Femoris

A significant muscular grouping, consisting of four dominant muscles located at the front of the thigh, crucial for extending the knee.

Ventrolateral Surface

Refers to the surface of an anatomical structure that is both toward the belly and the side.

Biceps Brachii

A large muscle in the upper arm responsible for flexion of the elbow and supination of the forearm.