   SEARCH HOME Math Central Quandaries & Queries  Question from Christina, a student: I'm having trouble solving for a second derivative for the following graphing question. f(x) = (X^2+2x+4)/2x using the quotient rule, I found: f'(x) = (x^2-4)/(2x^2) however, using the quotient rule again I can't seem to solve it (concavity): f'''(x)=[(2x)(2x^2)-(x^2-4)(4x)]/[(2x^2)^2] f''(x)=[(4x^3-(4x^3 -16x)]/4x^4 f''(x)=16x/4x^4 f''(x)=4/x^3 and making the equation equal to zero result in 0=4 which doesn't seem to make sense... Christina,

Your differentiation is fine and you found f''(x) = 4/x3. Then you attempted to solve f''(x) = 0 and ended with a statement that makes no sense, so there is no x so that f''(x) = 0. But the concavity might change at a place where f''(x) = 0 but also at a place where f''(x) is undefined. f''(x) = 4/x3 is undefined at x = 0 so if the concavity changes it does so at x = 0.

Penny     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.