Locate the absolute extrema of the given function (if any exist) over the indicated interval.

Question

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Locate the absolute extrema of the given function (if any exist) over the indicated interval.

f (x) = 3x²+6x-2

a) [-1,4]

b) [-1,4)

c) (-1,4]

d) (-1,4)

Answer 1

3x^2 +6x -2 is an upward opening parabola with vertex = absolute minimum = (-1,-5)= -5 when the x interval includes -1 indicated by the left bracket, [

for [-1,4]

f(-1)= -5 = absolute minimum = (-1,-5)

f(4)= 48+24-2= 70= absolute maximum= (4,70)

for [-1,4)

abs max DNE, abs min=-5

for (-1,4]

abs min DNE, abs max = 70

for (-1,4)

abs extrema DNE

f'(x) = 6x+6=0

solve for x

vertex =(-1,-5)

x =-1, f(-1)= y =-5

f=3x^2 +6x-2

complete the square:

= 3(x^2 +2x +1)-2-3

= 3(x+1)^2 -5

in vertex form

= a(x-h)^2 +k where (h,k)=vertex =(-1,5)

extrema do not exist DNE, when y approaches a value as a limit, but never reaches it