For the Function $f ( x ) = 2 x ^ { 3 } - 12 x ^ { 2 } + 2$

question 71

Multiple Choice

For the function $f ( x ) = 2 x ^ { 3 } - 12 x ^ { 2 } + 2$ : (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) \(x = 0,4\)
• (b) increasing: \(( - \infty , 0 ) \cup ( 4 , \infty )\); decreasing: \(( 0,4 )\)
• (c) relative max: \(f ( 0 ) = 2\); relative min: \(f ( 4 ) = - 62\)
• B) (a) \(x = 0,4\)
• (b) decreasing: \(( - \infty , 0 ) \cup ( 4 , \infty )\); increasing: \(( 0,4 )\)
• (c) relative min: \(f ( 0 ) = 2\); relative max: \(f ( 4 ) = - 62\)
• C) (a) \(x = 0,1\)
• (b) increasing: \(( - \infty , 0 ) \cup ( 1 , \infty )\); decreasing: \(( 0,1 )\)
• (c) relative max: \(f ( 0 ) = 2\); relative min: \(f ( 1 ) = - 8\)
• D) (a) \(x = 0,1\)
• (b) decreasing: \(( - \infty , 0 ) \cup ( 1 , \infty )\); increasing: \(( 0,1 )\)
• (c) relative min: \(f ( 0 ) = 2\); relative max: \(f ( 1 ) = - 8\)
• E) (a) \(x = 0,1\)
• (b) increasing: \(( - \infty , 0 ) \cup ( 1 , \infty )\); decreasing: \(( 0,1 )\)
• (c) relative \(\max : f ( 0 ) = 2\); no relative min.

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