Find a, b, c, and d such that the cubic function f(x) = ax^3 + bx^2 + cx + d satisfies the given conditions.

Question

Relative maximum: (3, 9)

Relative minimum: (5, 7)

Inflection point: (4, 8)

a =

b =

c =

d =

Answer 1

essentially we will have to create a system of equations to find our answer.

first and foremost lets plug in the given points as x and y leading to:

27a+9b+3c+d=9

64+16b+4c+d=8

125a+25b+5c+d=7

next we know that a minimum and maximum are at (3,9) and (5,7)

this means f'(x)=0 at those points

f'(x) = 3ax^2+2bx+c

meaning we get

75a+10b+c=0

and

27a+6b+c=0

now we can use these equations to solve a system of equations and get our variables. To do this we can input the equations into a matrix and use our row reduction function to get our values for a,b,c,d

doing this leads to a=1/2, b=-6, c=45/2, d=-18

to check to see if our point of inflection is (4,8) we find f''(x) which equals

24a+2b=0

and since 24(1/2)-2(6)=0 our values for a, b, c, and d check out