Question

Find a cubic function f(x)=ax^3+cx^2+d that has a local maximum value of 10 at -3 and a local minimum value of 6 at 0

what are
a
b
c

Answer 1

We have
\[f(x)=ax^3+cx^2+d\]
It has local maximum of 10 at x=-3 and local minimum of 6 at 0
so
\[10=-27a+9c+d\]
and
\[6=d\]
Wait a minute
Where is b in f(x)???

Answer 2

f(x)=ax^3+cx^2+d

f'(x) = 3ax^2+2cx
since it has minima at -3 and maxima at 0
f'(-3) = 0
f(-3) = 10
f'(0) = 0
f(0) = 6

3 variables 4 equations, should be solvable.