Give an example of a function, f(x), that has an inflection point at (1,4)

The zeroes of the 2nd derivative represents the inflection
points of a function.

The order of the function to be
differentiated is 3rd order.

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

We'll differentiate f(x) with respect to
x:

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

We'll differentiate with respect to x
again:

f"(x) = 6ax + 2b

We'll
cancel f"(x) = 0 knowing that the zero of f"(x) is x =
1.

6a + 2b = 0

3a + b =
0

b = -3a

We'll compute f'(1)
= 3a + 2b + c => f'(1) = b + c

We'll compute f(1) =
a + b + c + d

We also know that f(1) = 4 => a + b +
c + d = 4

Since the number of unknown coefficients is
larger than the number of possible equations, the function cannot be determined under
the circumstances.

Therefore, any polynomial
of 3rd order, at least: f(x) = ax^3 + bx^2 + cx + d, could have an inflection point
(1,4), under given circumstances.