Question

what are the maximum and minimum values of 2x3-9x2 +12x+15
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Answer 1

2x3−9x2−24x+15

dydx=6x2−18x−24

Let us determine the coordinates of the maxima and minima,

When dydx=0,

6x2−18x−24=0

x2−3x−4=0

Factor and solve,

(x−4)(x+1)=0

x=4or−1

When x=4,

y=2(4)3−9(4)2−24(4)+15

y=−97

(4,−97)

When x=−1

y=2(−1)3−9(−1)2−24(−1)+15

y=28

(−1,28)

Now, to determine the nature of these coordinates,

Find d2ydx2,

d2ydx2=12x−18

When x=4,

d2ydx2=12(4)−18

d2ydx2=30>0 ( minima )

When x=−1,

d2ydx2=12(−1)−18

d2ydx2=−30<0 ( maxima )

Therefore,

(4,−97) minima and (−1,28) maxima

Check:

graph{2x^3-9x^2-24x+15 [-20, 20, -120,120]}