Sketch a graph of a function with the following properties.

Question

For the following problems, make a sketch of the graph of a function that has the properties described.

1. f(3)=4; f'(3)=0; concave down for all x.
2. f(-1)=0; f'(x)< 0 for x < -1; f'(-1)=0; f'(x) > 0 for x > -1.
3. f(1)=2; f'(x) > 0 for x < 1; f'(1)=0; f'(x) > 0 for x > 1.
4. (-2, -1) and (2, 5) are on the graph; f'(-2)=0 and f'(2)=0; f''(x) > 0 for x < 0; f''(0)=0; f''(x) < 0 for x > 0.
5. (0, 5), (2, 1) and (4, -1) are on the graph; f'(0)=0 and f'(4)=0; f''(x) < 0 for x < 2; f''(2)=0; f''(x) > 0 for x > 2.
6. f(x) is defined only for x ≥ 0; (0, 0) and (5, 6) are on the graph; f'(x) > 0 for x ≥ 0; f''(x) < 0 for x < 5; f''(5)=0; f''(x) > 0 for x > 5.

Answer 1

Number 4 should look something like

and it should be a cubic function with a negative leading coefficient.

Answer 2

Here are examples that will work for the first 3 problems.

The key to this is to just dig in starting with the first bits of information and then sketching in as you go.

When it say f '(x) >0 then the graph will slant from lower left toward upper right. When it say f '(x) <0 then the graph will slant from upper left toward lower right. When it says f '(x) = 0 at a particular x-value, then you can draw a minimum, maximum or point of inflection there.

These rules might help:

1. If f ’ > 0 on (a, b), then f is increasing on (a, b)
2. If f ’ < 0 on (a, b), then f is decreasing on (a, b)
3. If f ’’ > 0 on (a, b), then f is concave up on (a, b)
4. If f’’ > 0 on (a, b), then f is concave down on (a, b)

Answer 3