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

1 Answer
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'(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#.

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