`f(x) = xsqrt(9 - x)` Find the points of inflection and discuss the concavity of the graph of the function.

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Chapter 3, 3.4 - Problem 22 - Calculus of a Single Variable (10th Edition, Ron Larson).
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embizze | High School Teacher | (Level 2) Educator Emeritus

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Find the inflection points and discuss the concavity of the function `f(x)=x*sqrt(9-x) `

The inflection points occur when the second derivative is zero (and changes sign.) A function's graph is concave up if the second derivative is positive, and concave down if the second derivative is negative.

Rewrite the function as:

` f(x)=x(9-x)^(1/2) `

Find the derivative using the product rule:

`f'(x)=(9-x)^(1/2)+x(1/2)(9-x)^(-1/2) `

`f''(x)=1/2(9-x)^(-1/2)+1/2(9-x)^(-1/2)+x/2(-1/2)(9-x)^(-3/2) `

`=(9-x)^(-1/2)-x/4(9-x)^(-3/2) `

`=(9-x)^(-3/2)(9-x-x/4) `

Setting the second derivative equal to zero yields:

x=9 or x=36/5.

9 is the right endpoint of the domain so it is not an inflection point. The sign of the second derivative is negative on the domain, so the function has no inflection points and is concave down on its domain.



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