Question

What are the points of inflection, if any, of #f(x)=x^4-5x^3+x^2 #?

Answer 1

`x=+-sqrt(5/2)`

Explanation:

To find points of inflection you need to find points where the slope is zero - at those points the slope is changing from positive to negative (decreasing), or from negative to positive (increasing). This means that the second derivative will be negative in the first case and positive in the second case.

`f'(x)=4x^3-15x^2+2x`

`f''(x)=12x^2-30`

This will change sign at the point where `x^2=30/12 = 5/2`

There are therefore two points of inflection, at `x=+-sqrt(5/2)`. Between these two values `f''(x)` is negative and elsewhere it is positive.