Question

For what values of x is `f(x)= (5x-x^3)/(2-x)` concave or convex?

Answer 1

First, find the point(s) where the first derivative is zero. Then determine whether the second derivative there is positive or negative.

Explanation:

A 'positive' second derivative value at an inflection point indicates a convex curve. A 'negative' one indicates a concave curve.

Weisstein, Eric W. "First Derivative Test." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/FirstDerivativeTest.html
Weisstein, Eric W. "Second Derivative Test." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/SecondDerivativeTest.html

REFERENCES:
Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 14, 1972.

Thomas, G. B. Jr. and Finney, R. L. "Maxima, Minima, and Saddle Points." §12.8 in Calculus and Analytic Geometry, 8th ed. Reading, MA: Addison-Wesley, pp. 881-891, 1992.