f72fa1…: “Up to the short tree”

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1 : Consider the function $f(x) = (\sage {p1f1}) + (\sage {p1f2}) + (\sage {p1f3}) + (\sage {p1f4})$ . What are the maximum number of relative extrema that $f(x)$ could have? $\answer [id=c]{\sage {p1ans1}}$ .

Remember, you don’t need to factor or graph the function, but you may need to simplify (combine like terms) the function to determine your answer. The answer is only asking for the maximum possible relative extrema, not how many local extrema the function actually has.

1.1 : What is the minimum number relative extrema that $f(x)$ could possibly have? $\answer [id=b]{\sage {p1ans2}}$

Remember, you don’t need to factor or graph the function, but you may need to simplify (combine like terms) the function to determine your answer. The answer is only asking for the minimum possible relative extrema, not how many local extrema the function actually has.

1.1.1 : Enter any number that could be a valid number of possible local extrema for $f(x)$ .

$\answer [id=a]{\sage {p1ans1}}$

This should allow you an opportunity to make sure you understand what possible numbers of relative extrema $f(x)$ can have. This should dynamically accept any valid answer, but it is still very experimental.

2 : Consider the function $f(x) = (\sage {p2f1}) + (\sage {p2f2}) + (\sage {p2f3}) + (\sage {p2f4})$ . What are the maximum number of relative extrema that $f(x)$ could have? $\answer [id=cc]{\sage {p2ans1}}$ .

2.1 : What is the minimum number relative extrema that $f(x)$ could possibly have? $\answer [id=bb]{\sage {p2ans2}}$

Remember, you don’t need to factor or graph the function, but you may need to simplify (combine like terms) the function to determine your answer. The answer is only asking for the minimum possible relative extrema, not how many local extrema the function actually has.

2.1.1 : Enter any number that could be a valid number of possible local extrema for $f(x)$ .

$\answer [id=aa]{\sage {p2ans1}}$

3 : Consider the function $f(x) = (\sage {p3f1}) + (\sage {p3f2}) + (\sage {p3f3}) + (\sage {p3f4})$ . What are the maximum number of relative extrema that $f(x)$ could have? $\answer [id=ccc]{\sage {p3ans1}}$ .

3.1 : What is the minimum number relative extrema that $f(x)$ could possibly have? $\answer [id=bbb]{\sage {p3ans2}}$

Remember, you don’t need to factor or graph the function, but you may need to simplify (combine like terms) the function to determine your answer. The answer is only asking for the minimum possible relative extrema, not how many local extrema the function actually has.

3.1.1 : Enter any number that could be a valid number of possible local extrema for $f(x)$ .

$\answer [id=aaa]{\sage {p3ans1}}$

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