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

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### 1 Answer

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.**