# Point of Inflection

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A curve y = f(x) has one of its points x = c as an inflection point

1) if f "(c)= 0 or is not defined and

2) if f "(x) changes sign as x increases through x = c.

The 2nd condition may be replaced by f "'(c ) $\neq$ 0 when f "'(c ) exists.

Thus, x = c is a point of inflection if f "(c) = 0 and F "'(c) $\neq$ 0

## Examples on Point of Inflection

**Example 1 :**Determine the point of inflection of f(x) = $x^{3}-6x^{2}+11 $

**Solution :**f(x) = $x^{3}-6x^{2}+11 $

f '(x) = $3x^{2}-12x $

f "(x) = 6x -12 To find the inflection point set f "(x) = 0

6x -12 = 0

6x = 12

x = 2

To find the y-coordinate of inflection point plug in x =2 in the given equation

f(x) = $x^{3}-6x^{2}+11 $

f(2) = $2^{3}-6(2)^{2}+11 $

= 8 - 24 + 11

f(2) = -5

**Inflection point (2,-5).**

**Example 2 :**Determine the point of inflection of f(x) = $x^{4}-6x^{2}+4 $

**Solution :**f(x) = $x^{4}-6x^{2}+4 $

f '(x) = $4x^{3}-12x $

f "(x) = $12x^{2}-12 $

To find the inflection point set f "(x) = 0

$12x^{2}-12 $ = 0

12($x^{2}$ -1)= 0

$x^{2}$ -1 = 0

$x^{2}$ = 1

x = $\pm$ 1

To find the y-coordinate of inflection point plug in x =1 and x = -1 in the given equation

f(x) = $x^{4}-6x^{2}+4

f(1) = $(1)^{4}-6(1)^{2}+4

= 1 - 6 + 4

**f(1) = -1**

f(x) = $x^{4}-6x^{2}+4

f(-1) = $(-1)^{4}-6(-1)^{2}+4

= 1 - 6 + 4

**f(1) = -1**

**Inflection points (1,-1) and (-1,-1).**

**12th grade math**

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