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.