Is #f(x)=x^3-x+e^(x-x^2) # concave or convex at #x=1 #?
It is convex.
The function will be convex if the second derivative is positive, and concave if it is negative.
Weisstein, Eric W. "Second Derivative Test." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/SecondDerivativeTest.html
The first derivative is:
The second derivative is:
Calculation detail steps here: