Is #f(x)=-3x^5-2x^4-6x^3+x-7# concave or convex at #x=5#?

1 Answer
Aug 16, 2017

convex (concave DOWNWARD)


It's a pretty straightforward polynomial, so taking the derivative is pretty easy. You can just derive the terms one by one, proceeding left to right:

#d/dx -3x^5 - 2x^4 - 6x^3 + x - 7 = -15x^4 - 8x^3 -18x^2 + 1#

And then, do it again:

#d/dx -15x^4 - 8x^3 -18x^2 + 1 = -60x^3 - 24x^2 - 36x #

You can plug in x = 5 and calculate if you want, but if you're pressed for time on an exam it's not really necessary. Each term in the second derivative is negative for x=5, so the second derivative is therefore negative at that point, so it's convex (or concave downward) at x = 5.