College algebra

Question

The function m(x)= −2x cubed − 3x squared + 252x + 4 has one local minimum and one local maximum. Graph the function on your calculator to estimate these local extrema and the intervals on which m is increasing and decreasing.

m(x) has a local maximum of _____ at x = 

m(x) has a local minimum of  _____ at x = 

m(x) is increasing on the interval(s):    

m(x) is decreasing on the interval(s):

Answer 1

f(x) = -2x^3 - 3x^2 + 252x + 4

take the derivative and set it = 0

f'(x) = -6x^2 -6x +252 = 0

divide by -6

x^2 +x - 42 = 0

(x-6)(x+7) =0

x-6 = 0 and x+7=0

x= 6 or -7

y = f(6) or f(-7) = -2(6)^3 -3(6)^2 + 252(6) + 4 = -432 -108 +1512 + 4 =976

or -2(-7)^3 -3(-7)^2 + (252)(-7) + 4 = 686-147 -1764 +4 = - 1221

the points (6,976) and (-7,-1221) are local max and minimum points

f(x) is increasing on the intervals (-infinity, -7) and (6, infinity)

f(x) is decreasing on the interval (-7,6)

slope = 0 at x=-7 and x=6

f"(6) = -12(-6) -12 >0 concave down

f"(-7) =-12(7) -12) <0 concave up