NOTE: This question is fairly sophisticated and is here to show you some of the complications that can occur. If you don't fully understand it now, don't worry!

1. x-intercepts

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

`=x^4(x-5)`

`=0`

when `x = 0`, `x = 5`

 

2. y-intercepts:

When `x = 0`, `y = 0`.

 

3. maxima and minima?

`(dy)/(dx)=5x^4-20x^3`

`=5x^3(x-4)`

`=0`

when `x = 0` or `x = 4`

So we have max or min at `(0,0)` and `(4,-256)`.

 

4. Second derivative:

`(d^2y)/(dx^2)=20x^3-60x^2`

Now `y” = 0` for `x = 0` so `(0,0)` is ???

Now `y” > 0` for `x = 4` so `(4,-256)` is a local MIN

 

5. Points of inflection

We now use the second derivative to find points of inflection:

`(d^2y)/(dx^2)=20x^3-60x^2`

`=20x^2(x-3)`

`=0`

when `x = 0` or `x = 3`

If `x < 0`, `y” < 0`: Local maximum shape

 

If `0 < x < 3`, `y” < 0`: Local maximum shape

 

There is no sign change, so at `x = 0`, there is NO point of inflection.

If `x > 3`, `y” > 0`: Local minimum shape

So the sign of `y”` has changed, so `(3,-162)` is a point of inflection.

Actually, at `x = 0`, we have a FLAT SPOT. It is not a local maximum, even though it may appear so.

So we are ready to sketch the curve:

Graph of polynomial showing max, min, point of inflection

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