Answer: By inspection, the x-intercept is at x = 4.
Next, find the y-intercepts. When x = 0, y = ≈ 2.52.
No asymptotes exist because there’s no place where the curve is undefined.
The first derivative is
Set it equal to zero.
This can never equal zero. But, at x = 4 the derivative is undefined, so this is a critical point. If you look
at the limit as x approaches 4 from both sides, you can see if there’s a cusp.
The curve has a cusp at (4, 0).
There were no other critical points. But, we can see that when x > 4, the derivative is positive and the
curve is rising; when x < 4 the derivative is negative, and the curve is falling.
The second derivative is
No value of x can set this equal to zero. In fact, the second derivative is negative at all values of x except
- Therefore, the graph is concave down everywhere.
Your graph should look like the following: