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Finding Points of Inflection










































Find the points of inflection for:

f(x)=x√100-x^2

An inflection point is a point where the curvature changes from convex to concave, or vice versa.

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Mar 05, 2012
Points of Inflection
by: Staff


Question:

f(x)=x√100-x^2


Answer:

I’m not quite sure what your function is.


A. f(x) = x√100 - x^2

Or

B. f(x) = x√(100 - x^2)


I’ll work it out both ways for you.


An inflection point is a point where the curvature changes from convex to concave, or vice versa.


To find the value(s) of x at the inflection point(s):

1) take the second derivative of the function
2) set the second derivative equal to 0
3) solve for x


A. f(x) = x√100 - x²


You can view a graph of the function f(x) = x√100 - x² by opening the following link:

(1) If your browser is Firefox, click the following link to VIEW the graph; or if your browser is Chrome, Internet Explorer, Opera, or Safari (2A) highlight and copy the link, then (2B) paste the link into your browser Address bar & press enter:

Use the Backspace key to return to this page

http://www.solving-math-problems.com/images/parabola-01-2012-03-05.png


As you can see, this is a standard parabola.

There are no inflection points.

We can verify this with the following calculations:


f(x) = x√100 - x^2

f(x) = 10x - x^2

f(x) = - x^2 + 10x


1st derivative

f’(x) = 10 - 2x


2nd derivative

f’(x) = 10 - 2x

f’’(x) = - 2


>>> For f(x) = x√100 - x^2, there is NO INFLECTION POINTS.




B. f(x) = x√(100 – x²)


You can view a graph of the function f(x) = x√(100 - x²) by opening the following link:

(1) If your browser is Firefox, click the following link to VIEW the graph; or if your browser is Chrome, Internet Explorer, Opera, or Safari (2A) highlight and copy the link, then (2B) paste the link into your browser Address bar & press enter:

Use the Backspace key to return to this page

http://www.solving-math-problems.com/images/x_sqrt_100-x2-graph-01-2012-03-05.png



f(x) = x√(100 - x^2)

The domain of this function is very limited. The only possible x values are:

-10 ≤ x ≤ 10


x cannot be less than -10 or greater than +10.


1st derivative

f(x) = x√(100 - x^2)

f’(x) = (-2x² + 100) / √[(10 - x) * √ (10 + x)]



2nd derivative

f’’(x) = -(2*x³-300*x)/[√(10-x) * √(x+10) * (x² - 100)]


-(2*x³-300*x)/[√(10-x) * √(x+10) * (x² - 100)] = 0

Solve for x

{-(2*x³-300*x)/[√(10-x) * √(x+10) * (x² - 100)]} * [√(10-x) * √(x+10) * (x² - 100)] = 0 * [√(10-x) * √(x+10) * (x² - 100)]

-(2*x³-300*x) * {[√(10-x) * √(x+10) * (x² - 100)] / [√(10-x) * √(x+10) * (x² - 100)]} = 0 * [√(10-x) * √(x+10) * (x² - 100)]

-(2*x³-300*x) * 1 = 0

-(2*x³-300*x) = 0

2*x³ - 300*x = 0

2x * (x² - 150) = 0

1st inflection point

2x = 0

x = 0


2nd and 3rd inflection points

x² - 150 = 0

x² = 150

x = ± √150

x = ± √150

x = ± 5√(6)

x = ± 12.2474487139159

Both of these inflection points are INVALID. The domain of x is restricted to: -10 ≤ x ≤ 10

>>> For f(x) = x√(100 – x²), there is only one inflection point:

x = 0


Thanks for writing.

Staff
www.solving-math-problems.com


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