Calculus of a Single Variable, Chapter 3, 3.4, Section 3.4, Problem 22

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.