Question

How do you find the inflection points of the graph of the function: `y=x^3-15x^2+33x+100`?

Answer 1

The inflection point is `(x,y)=(5,15)`.

Explanation:

If `f(x)=x^3-15x^2+33x+100`, then the first derivative is `f'(x)=3x^2-30x+33` and the second derivative is `f''(x)=6x-30=6(x-5)`. The second derivative changes sign (from negative to positive) as `x` increases through `x=5`, making this the first coordinate of the unique inflection point. In this case, the inflection point occurs where the graph of `y=f(x)` changes from concave down (frown-like) to concave up (smile-like).

The second coordinate of the inflection point is `f(5)=5^3-15*5^2+33*5+100`

`=125-375+165+100=15`.

The inflection point is therefore `(x,y)=(5,f(5))=(5,15)`