Inflexion point

A small lecture on the inflexion point.....
learn from it....

1. INFLEXION POINTS PRESENTED BY: SAAD SUFYAN AHMED ( G1F18BSCS0131)

2. Inflexion point An inflection point is a point on a curve at which the sign of the curvature (i.e., the concavity) changes. Inflection points may be stationary points, but are not local maxima or local minima. For example, for the curve y= plotted above, the point x=0 is an inflection point. The second derivative test is also useful. A necessary condition for x to be an inflection point is f“(x)=0  

3. Inflexion points

4. Example of Inflexion points Example 1: Find all points of inflection for the function f (x) = . We want to find where the second derivative changes sign, so first we need to find the second derivative. The first derivative is f '(x) = 3x² and the second derivative is f "(x) = 6x. The second derivative is never undefined, and the only root of the second derivative is x = 0. The sign of f " does change there, since f "(x) is negative for x < 0 and positive for x > 0.  

5. Example 1: Therefore x = 0 is an inflection point. Looking at the graph of f (x), this makes sense:

6. Example of points of inflexion Example 2: Find all inflection points for the function f (x) = x⁴. The first derivative is f '(x) = 4x³ and the second derivative is f "(x) = 12x² . The second derivative is never undefined, and the only root of the second derivative is x = 0. However, f "(x) is positive on both sides of x = 0, so the concavity of f is the same to the left and to the right of x = 0. This means x = 0 is not a point of inflection. .

7. Example 2: Since x = 0 Looking at the graph of f (x), this makes sense: only root of f "(x), the function f has no inflection points.