How do you find points of inflection using differentiation?

To find points of inflection, determine where the second derivative changes sign.

Points of inflection are where a curve changes concavity, from concave up to concave down or vice versa. To find these points, you first need to find the second derivative of the function. The second derivative, denoted as \( f''(x) \), gives us information about the curvature of the function.

Start by finding the first derivative \( f'(x) \) of the function \( f(x) \). This represents the slope of the tangent to the curve at any point. Next, differentiate \( f'(x) \) to get the second derivative \( f''(x) \).

Once you have \( f''(x) \), solve the equation \( f''(x) = 0 \) to find potential points of inflection. These are the x-values where the second derivative is zero, indicating a possible change in concavity. However, not all solutions to \( f''(x) = 0 \) are points of inflection.

To confirm a point of inflection, you need to check if the second derivative changes sign around these x-values. Pick values slightly less than and slightly greater than each potential point of inflection and substitute them into \( f''(x) \). If \( f''(x) \) changes from positive to negative or from negative to positive, then the function has a point of inflection at that x-value.

For example, if \( f''(x) \) changes from positive to negative, the curve changes from concave up to concave down, indicating a point of inflection. Similarly, if \( f''(x) \) changes from negative to positive, the curve changes from concave down to concave up.