And to find the points of inflection, we set d2y dx2 = 0. ⇒ d2y dx2 = 12x − 10 = 0 ⇒ x = 56. As expected, we have one more stationary point than point of inflection, and this time all our points are real. To determine the order of our stationary points, we calculate the second derivative at x = −13, 2. dy dx = 14 at x = 2 and dy dx.. Summary. A curve's inflection point is the point at which the curve's concavity changes. For a function f (x), f (x), its concavity can be measured by its second order derivative f'' (x). f ′′(x). When f''<0, f ′′ < 0, which means that the function's rate of change is decreasing, the function is concave down.

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Inflection points are found in a way similar to how we find extremum points. However, instead of looking for points where the derivative changes its sign, we are looking for points where the second derivative changes its sign. Let's find, for example, the inflection points of f ( x) = 1 2 x 4 + x 3 − 6 x 2 . The second derivative of f is f.. Figure 7. Identifying the behavior of the graph at an x-intercept by examining the multiplicity of the zero. The x -intercept \displaystyle x=-3 x = −3 is the solution of equation \displaystyle \left (x+3\right)=0 (x + 3) = 0. The graph passes directly through the x -intercept at \displaystyle x=-3 x = −3. The factor is linear (has a degree.