The point in a curve, where the curvature sign changes is called as the inflection point. It is also called as inflection point.
To find inflection point we need to integrate quadratic expression twice. Here are the steps to find out the inflection point.
Consider the quadratic expression f(x) = 2x3 + 2x2 + 1
Step: 1 (First Step of Integration) f(x) = 2x3 + 2x2 + 1 By integrating 2x3 + 2x2 + 1, we will get f'(x) = 6x2 + 4x
Step: 2 (Second Step of Integration) By integrating f'(x) = 6x2 + 4x, we will get f''(x) = 12x + 4
Step: 3 Equalize the integrated expression to 0 to find the value of x. Hence, f''(x) = 12x + 4 = 0 12x = -4 x = -1/3 or -0.3333
Step: 4 Substitute the x value to the given (input) expression f(x) = 2x3 + 2x2 + 1 f(x) = 2(-1/3)3 + 2(-1/3)2 + 1 f(x) = 1.1481
Therefore the inflection point for the given quadratic expression f(x) = 2x3 + 2x2 + 1 is (1.1481 at x = -0.3333)