Graph the function. Label all intercepts, critical points, and inflection points.

Answer

• Find dots.

The function *f* is undefined and has a vertical asymptote when *x* = 0. This function has no roots.

The derivative is

This is zero when *x* = 1. Since , we have a critical point at (1,*e*).

The second derivative is

Use the quadratic formula to see if *x*^{2}-2*x* + 2 has any roots.

*f*" is undefined at 0, so *f* has no inflection points.

• Find shapes.

We start with the numberline.

The function is positive when *x* is positive and negative when *x* is negative.

The derivative is negative when *x* < 1 and positive when *x* > 1.

Now for the sign of the second derivative.

Since the polynomial *x*^{2} - 2*x* + 2 opens upwards and never hits the *x*-axis, it must always be positive. Since *e*^{x} and *x*^{4} are also positive, the only factor of *f*" that isn't always positive is *x*:

This means *f*" is negative when *x* is negative, and positive when *x* is positive.

Now we can find the shapes:

• Play connect-the-dots.

The final graph looks like this: