9d6c4d…: “Up to the short tree”

$\newenvironment {prompt}{}{} \newcommand {\ungraded }[0]{} \newcommand {\HyperFirstAtBeginDocument }[0]{\AtBeginDocument }$

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 that local extrema must come in pairs. To list all the possible number of local extrema, start with the maximum number, and then subtract two at a time to make the list. For example, if a polynomial has a maximum of 7 local extrema, then we can create the list of possible local extrema by subtracting 2 at a time, so there can only be 7, 5, 3, or 1 local extrema. Thus the minimum local extrema in this case would be 1.

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? $\answer [id=bb]{\sage {p2ans2}}$

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? $\answer [id=bbb]{\sage {p3ans2}}$

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

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

Press...	...to do
left/right arrows	Move cursor
shift+left/right arrows	Select region
ctrl+a	Select all
ctrl+x/c/v	Cut/copy/paste
ctrl+z/y	Undo/redo
ctrl+left/right	Add entry to list or column to matrix
shift+ctrl+left/right	Add copy of current entry/column to to list/matrix
ctrl+up/down	Add row to matrix
shift+ctrl+up/down	Add copy of current row to matrix
ctrl+backspace	Delete current entry in list or column in matrix
ctrl+shift+backspace	Delete current row in matrix

Type...	...to get
norm	$\|\|\blue{[?]}\|\|$
text	$\text{\blue{[?]}}$
sym_name	$\backslash\texttt{\blue{[?]}}$
abs	$\left\|\blue{[?]}\right\|$
sqrt	$\sqrt{\blue{[?]}}$
paren	$\left(\blue{[?]}\right)$
floor	$\lfloor \blue{[?]} \rfloor$
factorial	$\blue{[?]}!$
exp	${\blue{[?]}}^{\blue{[?]}}$
sub	${\blue{[?]}}_{\blue{[?]}}$
frac	$\dfrac{\blue{[?]}}{\blue{[?]}}$
int	$\displaystyle\int{\blue{[?]}}d\blue{[?]}$
defi	$\displaystyle\int_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}d\blue{[?]}$
deriv	$\displaystyle\frac{d}{d\blue{[?]}}\blue{[?]}$
sum	$\displaystyle\sum_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}$
prod	$\displaystyle\prod_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}$
root	$\sqrt[\blue{[?]}]{\blue{[?]}}$
vec	$\left\langle \blue{[?]} \right\rangle$
mat	$\left(\begin{matrix} \blue{[?]} \end{matrix}\right)$
*	$\cdot$
infinity	$\infty$
arcsin	$\arcsin\left(\blue{[?]}\right)$
arccos	$\arccos\left(\blue{[?]}\right)$
arctan	$\arctan\left(\blue{[?]}\right)$
sin	$\sin\left(\blue{[?]}\right)$
cos	$\cos\left(\blue{[?]}\right)$
tan	$\tan\left(\blue{[?]}\right)$
sec	$\sec\left(\blue{[?]}\right)$
csc	$\csc\left(\blue{[?]}\right)$
cot	$\cot\left(\blue{[?]}\right)$
log	$\log\left(\blue{[?]}\right)$
ln	$\ln\left(\blue{[?]}\right)$
alpha	$\alpha$
beta	$\beta$
gamma	$\gamma$
delta	$\delta$
epsilon	$\epsilon$
zeta	$\zeta$
eta	$\eta$
theta	$\theta$
iota	$\iota$
kappa	$\kappa$
lambda	$\lambda$
mu	$\mu$
nu	$\nu$
xi	$\xi$
omicron	$\omicron$
pi	$\pi$
rho	$\rho$
sigma	$\sigma$
tau	$\tau$
upsilon	$\upsilon$
phi	$\phi$
chi	$\chi$
psi	$\psi$
omega	$\omega$
Gamma	$\Gamma$
Delta	$\Delta$
Theta	$\Theta$
Lambda	$\Lambda$
Xi	$\Xi$
Pi	$\Pi$
Sigma	$\Sigma$
Phi	$\Phi$
Psi	$\Psi$
Omega	$\Omega$

Controls

Symbols

Settings