$\text { For the function } f ( x ) = 4 x ^ { 3 } - 48 x ^ { 2 } + 6 \text { : }$

question 84

Multiple Choice

$\text { For the function } f ( x ) = 4 x ^ { 3 } - 48 x ^ { 2 } + 6 \text { : }$
(a) Find the critical numbers of $f$ (if any) ;
(b) Find the open intervals where the function is increasing or decreasing; and
(c) Apply the First Derivative Test to identify all relative extrema. Then use a graphing utility to confirm your results.

• A)
• (a) \(x = 0,2\)
• (b) increasing: \(( - \infty , 0 ) \cup ( 2 , \infty )\); decreasing: \(( 0,2 )\)
• (c) relative max: \(f ( 0 ) = 6\); relative min: \(f ( 2 ) = - 154\)
• B)
• (a) \(x = 0,2\)
• (b) decreasing: \(( - \infty , 0 ) \cup ( 2 , \infty )\); increasing: \(( 0,2 )\)
• (c) relative min: \(f ( 0 ) = 6\); relative \(\max : f ( 2 ) = - 154\)
• C)
• (a) \(x = 0,2\)
• (b) increasing: \(( - \infty , 0 ) \cup ( 2 , \infty )\); decreasing: \(( 0,2 )\)
• (c) relative max: \(f ( 0 ) = 6\); no relative min.
• D)
• (a) \(x = 0,8\)
• (b) increasing: \(( - \infty , 0 ) \cup ( 8 , \infty )\); decreasing: \(( 0,8 )\)
• (c) relative max: \(f ( 0 ) = 6\); relative min: \(f ( 8 ) = - 1018\)
• E)
• (a) \(x = 0,8\)
• (b) decreasing: \(( - \infty , 0 ) \cup ( 8 , \infty )\); increasing: \(( 0,8 )\)
• (c) relative min: \(f ( 0 ) = 6\); relative max: \(f ( 8 ) = - 1018\)

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Definitions:

Primary Reinforcer

An innately satisfying stimulus, such as food or comfort, that fulfills a basic biological need and does not require learning to be effective.

Reward

Any form of compensation, recognition, or benefit used to reinforce positive behavior and outcomes within an organization.

Desirable Behavior

Behavior that is seen as positive or beneficial within a given context, often encouraged by society or organizations.

Secondary Reinforcer

A stimulus that has become reinforcing by being associated with a primary reinforcer; it indirectly satisfies biological needs.