Introduction to Inflection Points AMBER DONG AB CALCULUS

+ Introduction to: Inflection Points AMBER DONG AB CALCULUS P 8 2009

+ An Inflection Point Is… n Defined as n A point at which a graph changes concavity. n Green dot = inflection point n Concavity: Concave UP n Concave DOWN n n An Inflection point occurs when concave up and concave down meet

+ Also Known As… n. A section of the graph of f (x) is considered to be concave up if its slope increases as x increases = the derivative increases as x increases. n = f”(x)> 0 {positive} n n. A section of the graph of f (x) is considered to be concave down if its slope decreases as x increases. =the derivative decreases as x increases. n = f''(x) < 0 {negative} n

+ Concave DOWN n This should make sense, because f''(x) > 0 means that f'(x) is increasing, and this is the definition of concave up. Inflection Point Concave UP n SO, using this logic we can conclude that an Inflection Point = n A number x in the domain of a function f such that f’’(x) = 0

+ Finding Inflection Points n FIRST: Find the second derivative of the function f(x) n F”(x) n Ex) f(x)=3 x 3 + x 2 n f’(x)=6 x 2 + 2 x n f”(x)=12 x + 2 n SECOND: n n Find the points where f”(x)=0 These points (where f”(x)=0) are the only possible candidates for inflection points Ex) 12 x + 2= 0 x = -1/6

+ Checking: Be Careful! n However, not all points where f”(x)=0 are inflection points! n Inflection points occur ONLY when the graph of f”(x) goes from positive to negative OR negative to positive n (below the x axis above the x axis OR above the x axis below the x axis)

+ Finding Inflection Points THIRD: n Graph n Look the equation f”(x) at the zeros, where the graph = 0 n When the curve crosses the x axis an inflection point occurs n Crosses: Curve passes from below to above, or above to below the x axis.