Follow these 16 steps for the following three functions.
1. Determine the domain of your function.
2. Find the x-intercepts (feel free to use a quadratic formula calculator on Google)
3. Find the end-behavior of your graph (hint: horizontal asymptotes).
4. Find f’(x).
5. Find the critical values for your function.
6. Create a first derivative (+/-) sign interval using critical values and a point of your choice in each
interval
7. Determine the intervals over which f is increasing.
8. Find f’’(x).
9. Evaluate the second derivative at your critical values (plug in the values from Step 5 into f’’(x)).
Determine if f has a local max or local min at each critical value. Explain how you interpreted the value of
the second derivative.
10. Find the zeros of the second derivative. These are possible points of inflection.
11. Create a second derivative (+/-) sign interval using your zeros of the second derivative as cut-off
values.
12. Over what intervals is the second derivative concave up? Concave down?
13. Does the graph have any points of inflection?
14. Now - make a single number line. Plot any critical values and inflection points. In each section of the
number line add an arrow to indicate if f is increasing or decreasing on this section. Then add a curve
upward or a curve downward to indicate if f is concave up or concave down on this section of the graph.
15. Sketch the graph of f using the work you have done in parts 1 – 14. Use the x-intercept values given in
number 2 above. Label maximum and minimum values with both x & y coordinates. Label any points of
inflection with x & y coordinates (round to 1 decimal place).
16. Check to see if you graph matches the graph of the function on Desmos. Fix your work as needed.
You don’t need to include this graph.
Function 1: f(x) = 5x3 + 4x2 – 2x
Function 2: f(x) = (5x + x2)e2x
Function 3: f(x) = x2ln(4x2)