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

Aug 20, 2017

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:

$f ' \left(x\right) = 3 {x}^{2} - {e}^{x} = 0$

The numerical solution is:

$x = - 0.459 , 0.910 , 3.733$

The second derivative gives the concavity at these points.

$f ' ' \left(x\right) = 6 x - {e}^{x}$

• $f ' ' \left(a\right) < 0$: concave down about $a$
• $f ' ' \left(a\right) > 0$: concave up about $a$
• $f ' ' \left(a\right) = 0$: inflection point (neither) about $a$

$f ' ' \left(- 0.459\right) = 6 \left(- 0.459\right) - {e}^{- 0.459} \approx - 3.386 < 0$
$f ' ' \left(0.910\right) = 6 \left(0.910\right) - {e}^{0.910} \approx 2.976 > 0$
$f ' ' \left(3.733\right) = 6 \left(3.733\right) - {e}^{3.733} \approx - 19.41 < 0$

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