Question

How do you determine the intervals on which function is concave up/down & find points of inflection for `y=4x^5-5x^4`?

Answer 1

Investigate the sign of the second derivative.

Explanation:

`y=4x^5-5x^4`

`y'=20x^4-20x^3 = 20(x^4-x^3)`

`y'' =20(4x^3-3x^2) = 20x^2(4x-3)`

`y''` is never undefined and `y''=0` at `x=0` and at `x=3/4`.

Because `20x^2` is always positive, the sign of `y''` is the same as the sign of `4x-3` (or build a sign table of sign diagram or whatever you have learned to call it, for `y''`).

`y''` is negative (so the graph of the function is concave down, for `x<3/4` and

`y''` is posttive (so the graph of the function is concave up, for `x > 3/4`

The curve is concave down on the interval `(-oo,3/4)` and
concave up on `(3/4,oo)`.

The int `(3/4, -81/128)` is the only inflection point.