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?

---

Definitions:

Per Capita Energy Use

The average amount of energy consumed per person in a given area or country.

Developed Countries

Nations with a high level of economic development, typically characterized by a high gross domestic product (GDP) per capita.

Leveled Off

A state where a variable, after a period of growth or decline, stabilizes and shows no significant change.

Biodiesel

A renewable, clean-burning diesel substitute produced from a variety of biomass sources, such as vegetable oils and animal fats.