   Chapter 3.3, Problem 73E

Chapter
Section
Textbook Problem

# Show that the function g ( x ) =   x |   x   | has an inflection point at (0, 0) but g ′ ′ ( 0 ) does not exist.

To determine

To show:

That the function gx=xx has an inflection point at (0, 0) but g''(0) does not exist

Explanation

1) Concept:

Concavity test:

If f"(x)>0 then the graph of f is concave upward

If f"(x)<0 then the graph of f is concave downward

2) Definition:

A point P on a curve y=fx is called an inflection point if f is continuous there and the curve changes from concave upward to concave downward or from concave downward to concave upward at P

3) Calculation:

Given that gx=xx

Use x=x2 in above equation

gx=x·x2

Differentiate g using product rule

g'x=x*x2'+x'x2

Using chain rule,

g'x=x*12x2*2x+x'x2

g'x=x2+x2

Simplify

g'x=2x2

Again differentiate

g''x</

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