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How do you find the maximum, minimum and inflection points and concavity for the function #f(x)=18x^3+5x^2-12x-17#?

1 Answer
Jul 17, 2018

Answer:

Below

Explanation:

#f(x)=18x^3+5x^2-12x-17#

#f'(x)=54x^2+10x-12#

#f''(x)=108x+10#

For maximum or minimum points, #f'(x)=0#

#54x^2+10x-12=0#
#27x^2+5x-6=0#
#x=(-5+-sqrt(25+648))/54#
#x=(-5+-sqrt673)/54#
#x=(-5+sqrt673)/54# or #x=(-5-sqrt673)/54#


To determine whether the point is maximum or minimum,
At #((-5+sqrt673)/54,-19.85)#,
#f''(x)=51.88>0#
Therefore, it is minimum and concave up at #((-5+sqrt673)/54,-19.85)#

At #((-5-sqrt673)/54,-0.57)#
#f''(x)=-51.88 <0#
Therefore, it is maximum and concave down at #((-5-sqrt673)/54,-0.57)#


For inflexion points, #f''(x)=0#

#108x+10=0#
#x=-10/108=-5/54#

Test #(-5/54,-15.86)#
#f''(0)=0+10=10>0#
#f''(-5/54)=0#
#f''(-1/2)=-44 <0#
Therefore, since there is a change in concavity, a point of inflexion occurs at #(-5/54,-15.86)#