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

Apr 26, 2018

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:
$3 {x}^{2} + \left(- 2 x + 1\right) {e}^{- {x}^{2} + x} - 1$

The second derivative is:
$6 x + {\left(2 x - 1\right)}^{2} {e}^{- {x}^{2} + x} - 2 {e}^{- {x}^{2} + x}$

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

At $x = 1$:
$6 + 2 {e}^{2} = 20.78$ positive value