How do you find the local maximum and minimum values of #g(x)=x^3+5x^2-17x-21#?

1 Answer

Local Maximum : #(-4.57259929569, 65.6705658828)#
Local Minimum : #(1.2392659623, -32.48538069)#

Explanation:

From the given equation
#y=x^3+5x^2-17x-21#

take the first derivative

#y'=d/dx(x^3)+d/dx(5x^2)+d/dx(-17x)+d/dx(-21)#

#y'=3x^2+10x-17#

Set #y'=0# then solve for x

#3x^2+10x-17=0#

#x=(-10+-sqrt((10)^2-4(6)(-17)))/(6)#

#x=(-10+-sqrt(304))/(6)#

to values for x:

#x_1=-4.57259929569#
and

#x_2=1.2392659623#

Solve for corresponding values of y using #y=x^3+5x^2-17x-21# and the points are

Local Maximum: #(-4.57259929569, 65.6705658828)#
Local Minimum : #(1.2392659623, -32.48538069)#

Kindly see the graph below for better view

Desmos.com

God bless....I hope the explanation is useful.