Question

Is `f(x)=x^3-x+e^(x-x^2)` concave or convex at `x=1`?

Answer 1

It is convex.

Explanation:

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:
`3x^2 + (-2x +1)e^(-x^2+x)-1`

The second derivative is:
`6x + (2x -1 )^2 e^(-x^2+x) -2e^(-x^2+x)`

Calculation detail steps here:
http://calculus-calculator.com/derivative/

At `x = 1`:
`6 + 2e^(2) = 20.78` positive value