Answer :
The inflection points are ( x = 0 ) and ( x = -8.4 ).
To find the critical points and inflection points of the function ( f(x) = x^5 + 14x^4 + 23), we need to follow these steps:
Find the derivative of the function: Take the derivative of ( f(x) ) with respect to ( x ) to find its derivative function ( f'(x) ).
( f'(x) = 5x^4 + 56x^3 )
Set the derivative equal to zero and solve for ( x ): Set ( f'(x) = 0 ) and solve for ( x ) to find the critical points.
( 5x^4 + 56x^3 = 0 )
Factoring out common terms, we have:
( x^3(5x + 56) = 0 )
Setting each factor equal to zero gives us two possibilities:
( x^3 = 0 ) or ( 5x + 56 = 0 )
Solving these equations individually, we find:
( x = 0 ) (triple root)
( x = -\frac{56}{5} )
Therefore, the critical points are ( x = 0 ) and ( x = -\frac{56}{5} ).
Find the second derivative: Take the derivative of ( f'(x) ) to find its second derivative ( f''(x) ).
( f''(x) = 20x^3 + 168x^2 )
Determine the inflection points: To find the inflection points, we need to find the values of ( x ) where ( f''(x) = 0 ) or does not exist.
( 20x^3 + 168x^2 = 0 )
Factoring out common terms, we have:
( x^2(20x + 168) = 0 )
Setting each factor equal to zero gives us two possibilities:
( x^2 = 0 ) or ( 20x + 168 = 0 )
Solving these equations individually, we find:
( x = 0 )
( x = -\frac{168}{20} = -8.4 )
Therefore, the inflection points are ( x = 0 ) and ( x = -8.4 ).
In summary, the critical points of the function ( f(x) = x^5 + 14x^4 + 23 ) are ( x = 0 ) and ( x = -\frac{56}{5} ), and the inflection points are ( x = 0 ) and ( x = -8.4 ).
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