How do you find all points of inflection #f(x)=x^3-12x#?

An inflection point, also known as a point of inflection, is a point on the function's graph where the concavity changes. Since the sign of the second derivative indicates the concavity, an inflection point can also be defined as a point on the function's graph where the sign of the second derivative changes.

To find all points of inflection of $f(x) = x^3 - 12x$, follow these steps:

1. Find the second derivative of $f(x)$, denoted as $f''(x)$.
2. Set $f''(x) = 0$ and solve for $x$. These values of $x$ represent the potential points of inflection.
3. Determine the concavity of the function $f(x)$ around each potential point of inflection by considering the sign of the second derivative $f''(x)$ in the intervals determined by the critical points.
4. Confirm that the concavity changes at each potential point of inflection.

Let's go through the steps:

1. Find the first derivative: $f'(x) = 3x^2 - 12$

2. Find the second derivative: $f''(x) = 6x$

3. Set $f''(x) = 0$ and solve for $x$: $6x = 0$ $x = 0$

4. Now, determine the concavity around the point $x = 0$:

   • For $x < 0$, choose $x = -1$:
     • $f''(-1) = 6(-1) = -6$, so concave down.
   • For $x > 0$, choose $x = 1$:
     • $f''(1) = 6(1) = 6$, so concave up.

5. Since the concavity changes from concave down to concave up at $x = 0$, $x = 0$ is a point of inflection.

Therefore, the only point of inflection of $f(x) = x^3 - 12x$ is at $x = 0$.