These observations lead to a generalization for any function f(x) that has a derivative on an interval I:
2) If f '(x) < 0 on an interval I, then the graph of f(x) falls as x increases.
3) If f '(c) = 0, then the graph of f(x) has a horizontal tangent at x = c. The function may have a local maximum or minimum value, or a point of inflection.
2) To be a maximum point, the graph must change direction from increasing to decreasing.
3) To be an inflection point, the graph doesn't change direction. In the above example ( one in middle) it is increasing before the f '(c) = 0 and it is still increasing after. You can also have one with the graph decreasing on both sides.
Substitute these values into the original function to find the y values of the critical points. The points are (0, -4) and (-2, 0)
f '(x) | |
x < -2 | + |
x = -2 | 0 |
-2 < x < 0 | - |
x = 0 | 0 |
x > 0 | + |
c) Find the zeros of the original function. These are the x-intercepts. You can use synthetic division and factoring to find the zeros! They are (-2, 0) (double root) and (1, 0)
d) Find the y-intercept. This is the constant of the original function. (0, -4)
e) Now take the limit as x goes to both infinities of the original function.
f) Now put all these together and graph the function!
f '(x) | |
x < -2 | - |
x = -2 | 0 |
-2 < x < 0 | + |
x = 0 | 0 |
0 < x < 2 | - |
x = 2 | 0 |
x > 2 | + |
d) The y intercept is (0, 9)
e) Find the limits on infinity.
f) Put these together and graph the function!