Determine the Maximum Number of Turning Points of F $$f ( x ) = x ^ { 2 } ( x + 2 )$$

question 81

Short Answer

Determine the maximum number of turning points of f.
- $$f ( x ) = x ^ { 2 } ( x + 2 )$$
Answer: (a) For large values of $| x |$ , the graph of $f ( x )$ will resemble the graph of $y = x ^ { 3 }$ .
(b) $y$ -intercept: $( 0,0 )$ , $x$ -intercepts: $( 0,0 )$ and $( - 2,0 )$
(c) The graph of $f$ crosses the $x$ -axis at $( - 2,0 )$ and touches the $x$ -axis at $( 0,0 )$ .
(e) Local minimum at $( 0,0 )$ , Local maximum at $( - 1.33,1.19 )$
(f)

(g) Domain of f: all real numbers; range of f: all real numbers
(h) f is increasing on (-, -1.33) and (0, ); f is decreasing on (-1.33, 0)

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Definitions:

Inner Motivation

The drive to engage in an activity for its own sake and personal rewards, rather than for some external rewards or pressures.

Algorithms

A set of instructions designed to perform a specific task or solve a particular problem.

Convergent Thinking

A thought process aimed at figuring out the most effective answer to a problem, utilizing logic and analysis.

Divergent Thinking

A cognitive strategy employed to develop original concepts by investigating a wide range of potential solutions.