Question

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

Answer 1

I never really call it convex or concave... and instead, concave up or down is easier.

• concave down about `x = -0.459, 3.733`
• concave up about `x = 0.910`

---

Setting the first derivative to zero gives the points where the function has a zero slope, i.e. the minima or maxima:

`f'(x) = 3x^2 - e^x = 0`

The numerical solution is:

`x = -0.459, 0.910, 3.733`

The second derivative gives the concavity at these points.

`f''(x) = 6x - e^x`

• `f''(a) < 0`: concave down about `a`
• `f''(a) > 0`: concave up about `a`
• `f''(a) = 0`: inflection point (neither) about `a`

`f''(-0.459) = 6(-0.459) - e^(-0.459) ~~ -3.386 < 0`
`f''(0.910) = 6(0.910) - e^(0.910) ~~ 2.976 > 0`
`f''(3.733) = 6(3.733) - e^(3.733) ~~ -19.41 < 0`

So the function is concave down about `x = -0.459, 3.733`, and concave up about `x = 0.910`.