Let

f$($x$)=$ x$4($x $1)3.$

$($a$)$

Find the critical numbers of the function f$.($Enter your answers from smallest to largest.$)$

smallest value

x$1$

$=$ Way to go$!$

x$2$

$=$ Good!largest value

x$3$

$=$ Amazing job!

$($b$)$

What does the Second Derivative Test tell you about the behavior of f at these critical numbers?

At

x$1$

the second derivative test $---$Select$---$ indicates a local minimum indicates a local maximum indicates neither a minimum nor a maximum is inconclusive $.$

At

x$2$

the second derivative test $---$Select$---$ indicates a local minimum indicates a local maximum indicates neither a minimum nor a maximum is inconclusive $.$

At

x$3$

the second derivative test $---$Select$---$ indicates a local minimum indicates a local maximum indicates neither a minimum nor a maximum is inconclusive $.$

$($c$)$

What does the First Derivative Test tell you? Note what the First Derivative Test tells you that Second Derivative Test does not.

At

x$1$

the first derivative test $---$Select$---$ indicates a local minimum indicates a local maximum indicates neither a minimum nor a maximum is inconclusive $.$

At

x$2$

the first derivative test $---$Select$---$ indicates a local minimum indicates a local maximum indicates neither a minimum nor a maximum is inconclusive $.$

At

x$3$

the first derivative test $---$Select$---$ indicates a local minimum indicates a local maximum indicates neither a minimum nor a maximum is inconclusive