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

##### 1 Answer

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