# How do you find the inflection points of #f(x)= 12x^5+45x^4-80x^3+6#?

##### 1 Answer

Maximum: x=-4

Inflection: x=0

Minimum: x=1

#### Explanation:

In general, this is a hard problem for a quintic equation. But this particular quintic has some missing low order terms that help us a great deal.

To find the inflection and turning points of

Now this is a quartic equation. A full "solution by radicals" of the quartic is known, but it is long and complex (and painful to perform!). Fortunately, we can factorise this equation:

So

To categorise these consider the second derivative:

So -4 is a maximum of the function and +1 is a minimum. To categorise 0 we need to take another derivative.

Thus 0 is a point of inflection.

We want to sanity check our answer by comparing to the graph of the function, but the maximum at -4 is off the scale compared to the other points of interest. Plot it twice, once zoomed in, once zoomed out:

graph{12x^5+45x^4-80x^3+6 [-5, 2, -20, 20]}

graph{12x^5+45x^4-80x^3+6 [-5, 2, -1000, 5000]}