STATEMENT OF PROBLEM

We are going to investigate the relative extreme values of the sum of two functions: x^2 and A Cos[x], where the parameter A will vary.

1. Plot the graph of f[x] = x^2 on [-10,10].

2. Plot the graph of g[x] = Cos[x] on [-2 Pi, 2 Pi].

3. Predict and sketch the shape of the graph of h[x]=x^2 + Cos[x] on [-10,10].

4. Plot the graph of h on [-10,10] to verify your conjecture. What do you observe?

5. Find all values of x for which h has relative maximum or minimum values. Explain your results.

6. Predict the shape of the graph of h7[x]=x^2 + 7 Cos[x] on [-10,10].

7. Plot the graph of h7 on [-10,10] to verify your conjecture. What do you observe?

8. Find all values of x on [-10,10] for which h7 has relative maximum or minimum values.

9. Consider the function hA[x]=x^2 + A cos x on [-10,10] where A is a constant. For what values of A will hA have relative extreme values? How many relative extreme values can we expect? Justify your answer.