All local maximums and minimums on a function’s graph are called local extrema.
#f(x) =2x^3-3x^2-36x-3 or f^'(x) = 6x^2 -6x- 36#
At critical points #f^'(x)=0 :. 6(x^2-x-6)=0 # or
#x^2-x-6=0 or (x-3)(x+2) =0 # ;Critical points are
#x=-2, x=3 # when #x=-2 , f(x)= 2(-2)^3-3(-2)^2-36(-2)-3=41#
when #x=3 , f(x)= 2* 3^3-3*3^2-36*3-3= -84# . To test
local maximum or minimum we will examine sign change
at three positions #x = -3 , x =0 ,x=4 # for
#f^'(x) = 6x^2 -6x- 36 ; f^'( -3) = 36 # (increasing) ,
#f^'(0) = -36 # (decreasing) and #f^'(4) = 36 # (increasing)
#x=-2# is local maximum and #x=3 # is local minimum
Hence local maximam is at # (-2,41)# and local minimum
is at # (3,-84)#
graph{2x^3-3x^2-36x-3 [-157, 157, -78.5, 78.5]}
