Summary
Highlights
An inflection point occurs when the second derivative is zero, and crucially, the concavity of the function changes (from concave up to down, or vice versa). These points typically lie between relative extrema.
Concavity describes the curvature of a graph. A function is concave up when its second derivative is positive and its first derivative is increasing. It's concave down when the second derivative is negative and the first derivative is decreasing.
To find inflection points and concavity, first, calculate the first and second derivatives. For f(x) = x^3 - 9x^2 + 7x, the first derivative is 3x^2 - 18x + 7, and the second derivative is 6x - 18. Setting the second derivative to zero gives x = 3 as a potential inflection point.
Using a sign chart for the second derivative (6x - 18), we find that for x < 3, the function is concave down (f''(x) < 0), and for x > 3, it's concave up (f''(x) > 0). Since concavity changes at x = 3, it's an inflection point. Plugging x = 3 into the original function gives f(3) = -33, so the inflection point is (3, -33).
For f(x) = x^4 + 4x^3 + 1, the first derivative is 4x^3 + 12x^2, and the second derivative is 12x^2 + 24x. Factoring the second derivative as 12x(x + 2) and setting it to zero gives potential inflection points at x = 0 and x = -2.
A sign chart for f''(x) = 12x(x + 2) reveals that the function is concave up for x < -2, concave down for -2 < x < 0, and concave up for x > 0. Since concavity changes at both x = -2 and x = 0, both are inflection points. The corresponding y-coordinates are f(0) = 1 and f(-2) = -15, leading to inflection points at (0, 1) and (-2, -15).
It's crucial that the concavity must change for a point where the second derivative is zero to be considered an inflection point. If the sign of the second derivative doesn't change across that point, it's not an inflection point, even if f''(x) = 0.