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