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Algebra
Question

1) \( f ( x ) = 2 + 2 x ^ { 2 } - x ^ { 4 } \) a) ...

1) \( f ( x ) = 2 + 2 x ^ { 2 } - x ^ { 4 } \) 

a) Find \( f ^ { \prime } ( x ) \) 

b) Find the critical points. 

c) Construct a sign chart for the first derivative. Then determine the intervals where \( f ( x ) \) is increasing and the intervals where \( f ( x ) \) is decreasing 

d) Classify each critical point as a relative maximum/relative minimum or neither. 

e) Find \( f ^ { \prime \prime } ( x ) \) 

f) Construct a sign chart for the second derivative. Then determine the intervals where \( f ( x ) \) 

is concave up and the intervals where \( f ( x ) \) is concave down. 

g) Find the inflection point(s). 

h) Find the inflection point(s). 

Answer

a) 

\(f ^ { \prime } ( x ) = 2 + 2 x ^ { 2 } - x ^ { 4 } : \quad f ( x ) = 2 x + \frac { 2 x ^ { 3 } } { 3 } - \frac { x ^ { 5 } } { 5 } + c _ { 1 }\)

 

b) 

Critical Points of \( 2 + 2 x ^ { 2 } - x ^ { 4 } : \quad x = - 1 , x = 0 , x = 1\) 

 

e)

\(f ^ { \prime \prime } ( x ) = 2 + 2 x ^ { 2 } - x ^ { 4 } : \quad f ( x ) = x ^ { 2 } + \frac { x ^ { 4 } } { 6 } - \frac { x ^ { 6 } } { 30 } + c _ { 1 } x + c _ { 2 }\)

 

g) 

Inflection Points of \( 2 + 2 x ^ { 2 } - x ^ { 4 } : ( - \sqrt { \frac { 1 } { 3 } } , \frac { 23 } { 9 } ) : ( \sqrt { \frac { 1 } { 3 } } , \frac { 23 } { 9 } )\) 

 

 

 

 

Solution
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