Calculus

An inflection point of a graph is a point where the curve changes concavity. In other words, it is the point where the curvature of the graph changes from positive to negative or vice versa. For a polynomial function, an inflection point occurs when the second derivative of the function changes sign. To solve for the points of inflection, we set the second derivative equal to 0 and solve for x: f''(x)= 60x^3 + 180x^2 - 1080x = 0 D = b^2 - 4ac D = (180)^2 - 4(60)(-1080) = 324,000 x = (-180 +√324,000)/(2(60)) x ≈ 22.14 x1 = 22.14 To find the second solution, we substitute -b for b in the quadratic formula: x = (-180 - √324,000)/(2(60)) x ≈ -20.71 x2 = -20.71 The two points of inflection for the given function are x1 = 22.14 and x2 = -20.71.