Question

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

Answer 1

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.