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

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An inflection point of a function f(x) is the point where the curvature changes sign. When that happens f''(x) = 0.

Here an example of a function is required that has a point of inflection at (1, 4). Let the function be f(x).

f''(1) = 0

Let f''(x) = 6x - 6

f'(x) = 3x^2 - 6x + C

f(x) = x^3 - 3x^2 + C1*x + C2

f(1) = 4

=> 1^3 - 3*1 + C1*1 + C2 = 4

Let C2 = 0

=> C1 = 6

**An example of a function that has an inflection point at (1,4) is ****f(x) = x^3 - 3x^2 + 6x**

Hm if possible, could i have another expert answer? Its on my math review sheet and i wake to make sure :%

The function has an inflection point when the 2nd derivative is cancelling out.

We'll have to choose a function of 3rd order, to be differentiated twice.

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

We'll differentiate with respect to x:

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

We'll differentiate with respect to x again:

f"(x) = 6ax + 2b

We'll put f"(x) = 0 for x = 1

6a + 2b = 0

3a + b = 0

b = -3a

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

f'(1) = b + c

f(1) = a + b + c + d

But f(1) = 4

a + b + c + d = 4

To determine the function f(x), it would be necessary to provide another constraint concerning the 1st derivative, otherwise, the coefficients cannot be determined under the circumstances.

**But, you have to remember that the function is a polynomial of 3rd order, at least: f(x) = ax^3 + bx^2 + cx + d.**