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 x²-2x + 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 x² - 2x + 2 opens upwards and never hits the x-axis, it must always be positive. Since e^x and x⁴ 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: