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:

If 0 < x < 3, y” < 0:

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

If x > 3, y” > 0:

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:

The following points are indicated with dots:

x-intercepts (0,0) (and "flat spot") and (5,0) (green dots)

Point of inflection (3,-162) ("plus" sign)

Local minimum (4,-256) (magenta dot)