Learn how the second derivative of a function is used in order to find the function's inflection points. P Point of inflection . If P(c, f(x))is a point the curve y= f (x) such that f ‘() , Math video on how to determine intervals of concavity and find inflection points of a polynomial by performing the second derivative test. A positive second derivative means a function is concave up, and a negative second derivative means the function is concave down. The inflection point and the concavity can be discussed with the help of second derivative of the function. Concavity, convexity and points of inflexion Submitted By ... to concavity in passing through the point . And where the concavity switches from up to down or down to up (like at A and B), you have an inflection point, and the second derivative there will (usually) be zero. Find the intervals of concavity and the inflection points of f(x) = –2x 3 + 6x 2 – 10x + 5. These inflection points are places where the second derivative is zero, and the function changes from concave up to concave down or vice versa. Determining concavity of intervals and finding points of inflection: algebraic. Criteria for Concavity , Convexity and Inflexion Theorem. This is where the second derivative comes into play. If the concavity changes from up to down at \(x=a\), \(f''\) changes from positive to the left of \(a\) to negative to the right of \(a\), and usually \(f''(a)=0\). Problem 3. Of particular interest are points at which the concavity changes from up to down or down to up; such points are called inflection points. Practice questions. If the graph of flies above all of its tangents on an interval I, then it is called concave upward (convex downward) on I. Inflection points are points on the graph where the concavity changes. f '(x) = 16 x 3 - 3 x 2 Definition If f is continuous ata and f changes concavity ata, the point⎛ ⎝a,f(a)⎞ ⎠is aninflection point of f. Figure 4.35 Since f″(x)>0for x