Find the cubic function f(x)=ax3+bx2+cx+d that has a local maximum value of 3 at x=−2 and a local minimum value of 0 at x=1.

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Beverly Smith · Accepted Answer

Step 1  Find f

GaceCoect5v · Accepted Answer

f′(x)=3ax2+2bx+cf′(−2)=012a−4b+c=0(1)f′(1)=03a+2b+c=0(2)1−2⇒9a−6b=0or 3a=2balso (-2,3) lies on original−8a+4b−2c+d=3and (1,0) lies on ita+b+c+d=0subtract those two equations9a−3b+3c=−3,or3a−b+c=−1, (3)2−3:3b=1b=13, but a=2b3=(23)(13)=29in 33(29)−13+c=−1c=−43in a+b+c+d=029+13−43+d=0d=79so f(x)=(29)x3+(13)x2−(43)x+79

user_27qwe · Accepted Answer

f(−2)=−8a+4b−2c+d=3(equation 1)f(1)=a+b+c+d=0(equation 2)Subtracting equation 2 from equation 1,−9a+3b−3c=33a−b+c=−1(equation 3)f′(x)=3ax2+2bx+cf′(−2)=12a−4b+c=0(equation 4)f′(1)=3a+2b+c=0(equation 5)Subtracting equation 5 from equation 4,9a−6b=03a−2b=02b=3ab=1.5a(equation 6)Substitute equation 6 into equation 53a+2(1.5a)+c=06a+c=0c=−6a(equation 7)Substitute equations 6 and 7 into equation 1−8a+4(1.5a)−2(−6a)+d=3−8a+6a+12a+d=310a+d=3d=−10a+3(equation 8)Substitute equations 6, 7, and 8 into (equation 2)a+1.5a−6a+(−10a+3)=02.5a−16a+3=0−13.5a+3=0a=−3−13.5a=−627a=−29(equation 9)Substitute equation 9 into 6, 7, 8b=(1.5)⋅(−29)b=−39b=−13(equation 10)c=−6(−29)c=129c=43(equation 11)d=−10(−19)+3d=209+3d=2 and 29+3d=5 and 29(equation 12)f(x)=(−29)x3−(13)x2+(43)x+5 and 29

Find the cubic function f(x)=ax^{3}+bx^{2}+cx+d that has a local maximum

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