How do you find the inflection points for the function f(x)=xsqrt(5-x)f(x)=x5x?

1 Answer
Sep 12, 2014

Since f''(x) is always negative in the domain of f, the graph of f is always concave downward; therefore, there is no inflection point.

Let us first find the domain of f.
Since the expression inside the square-root cannot be negative,
5-x ge 0 Rightarrow 5 ge x,
so the domain of f is (-infty,5].

By Product Rule,
f'(x)=1cdot sqrt{5-x}+xcdot{-1}/{2sqrt{5-x}}={10-3x}/{2sqrt{5-x}}

By Quotient Rule,
f''(x)={-3cdot2sqrt{5-x}-(10-3x)cdot{-1}/{sqrt{5-x}}}/{4(5-x)}
={3x-20}/{4(5-x)^{3/2}}<0 on (-infty,5)

Hence, f is always concave downward.