local man, local min, inflection point problem
find a,b,c, and d such that the cubic equation f(x)=ax^3+bx^2+cx+d has a local maximum at (3,3); a local minimum at (5,1); and an inflection point at (4,2).
f'(x)=3ax^2+2bx+c
f''(x)=6ax+2b
local min and local max f'(x)=0 and inflection point f''(x)=0
so plugging in x=3 and x=5 for max and min into f'(x) and setting it equal to 0 i get
27a+6b+c=0
75a+10b+c=0
plugging in x=4 for f''(x) and setting it equal to 0 i get
24a+2b=0
also i know i have to plug in x=3, x=5, and x=4 into the original equation while setting each to 3, 1, and 2 to get
27a+9b+3c+d=3
125a+25b+5c+d=1
64a+16b+4c+d=2
but i know have a system of 6 equations with 4 unknowns and i know this system wont give me the values i am looking for since i have more equations than unknowns
however my professor did mention that i could get rid of 2 of the equations so ill have 4 equations with 4 unknowns, the reason why i am stuck up to this point is that i do not know which two equations to get rid of
i want to choose the the two equations dealing with the inflection point, that is
64a+16b+4c+d=2
24a+2b=0
but the only reason i can think of for choosing these two is because since there is a max at x=3 and min at x=5 then the graph has to increase hit the max at x=3 then decrease till it hits the min, so in between x=3 and x=5 the graph has to change concavity, and the question already says this happens at x=4 so we can just ignore those 2 equations, however im not too sure of my reasoning there haha
also could i just get rid of any 2 equations and still get the right answer?
please help and sorry for making this such a long post, just wanted anyone who sees my post what ive done so far and more
thanks in advance
