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

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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